Variable speed drive for the sensorless PWM control of an AC motor by exploiting PWM-induced artefacts

ABSTRACT

A variable speed drive comprises an output for delivering a drive voltage to an electric motor, power inverter, a drive controller, and a current sensor. The drive controller includes a PWM generator, a control law module, and a state variable estimator for estimating a state variable of the electric motor. The module computes a target voltage signal based on state variable estimates provided by the estimator and outputs the target voltage signal to the PWM generator. The generator approximates the target voltage signal with a PWM control signal, controls the inverter using the control signal, computes, based on the deviation between the control signal and the target voltage signal, an estimation support signal, and outputs the estimation support signal to the estimator. The estimator estimates a state variable of the motor based on the estimation support signal and the drive current, and outputs the estimate to the module.

The present application claims priority to U.S. provisional utilitypatent application Ser. No. 62/905,663 filed on Sep. 25, 2019 and toEuropean application serial no. 20305748.4 filed on Jul. 2, 2020, whichare both incorporated herewith in their entirety.

TECHNICAL FIELD

This disclosure generally pertains to the field of motor control. Moreprecisely, it relates to Variable Speed Drives (VSDs), which are used tocontrol the operation of Alternating Current (AC) electric motors. Thefocus is on VSDs that rely on Pulse-Width Modulation (PWM) to havecontinuous control over the speed of the controlled AC motor.

VSDs are typically used as industrial drives in factories,installations, HVAC systems and the like to control e.g. the position,speed and/or torque of an electric motor that is dedicated to aparticular task, such as e.g. the operation of a fan or the hoisting ofa load.

BACKGROUND ART

In order to perform a closed-loop control of an AC motor, a VSD needs tohave real-time information on the operating status of the AC motor. Thisinformation might for example be the instantaneous angular positionand/or angular velocity of the motor's rotor.

The VSD may obtain this information from dedicated sensors, which arearranged on the motor and monitor the motor's operating status. However,fitting an electric motor with such sensors adds to the complexity andsize of the whole drive assembly. The required sensors and sensorcabling also increase the price and reduce the reliability.

This is why so-called “sensorless” VSDs have become more and morecommon. In these sensorless VSDs, the motor's operating status isestimated on the basis of measurements of the motor's drive current. Noadditional external sensors are used. In order to improve theestimations, in particular when the motor operates at low velocity, itis a standard procedure to inject an external high-frequency probingsignal into the motor's drive voltage.

The article “Adding virtual measurements by signal injection” by PascalCombes et al., published in 2016 in the proceedings of the 2016 AmericanControl Conference on pages 999 ff., conceptualizes and generalizes thesignal injection technique for the sensorless control of electric motorsat low speed.

Signal injection is an effective method, but it comes at a price: theripple it creates may in practice yield unpleasant acoustic noise andexcite unmodeled dynamics. In particular, in the very common situationwhere the electric motor is fed by a PWM inverter, the frequency of theinjected probing signal may not be as high as desired so as not tointerfere with the PWM (typically, it cannot exceed 500 Hz in anindustrial drive with a 4 kHz-PWM frequency).

The article “A novel approach for sensorless control of PM machines downto zero speed without signal injection or special PWM technique” by C.Wang et al., IEEE Transactions on Power Electronics, Vol. 19, No. 6,November 2004, pages 1601 ff., proposes to measure the phase currentripples induced by conventional PWM to derive the rotor position andspeed of a PWM-controlled electric motor. However, this approach isbased on the measurement of current derivatives, which requiresspecialized current sensors and is sensitive to noise.

SUMMARY

In view of the above, it is an object of the present disclosure toprovide a PWM-based variable speed drive with an improved sensorless ACmotor control without signal injection.

According to the present disclosure, this object is achieved with avariable speed drive for the closed loop control of the operation of anAC electric motor based on a given control law, the variable speed drivecomprising:

-   -   an output terminal for delivering a controlled alternating drive        voltage to the controlled AC electric motor;    -   a solid-state power inverter for generating the drive voltage;    -   a drive controller for controlling the generation of the drive        voltage by the power inverter; and    -   a drive current sensing device for measuring the instantaneous        intensity of the drive current taken up by the controlled AC        electric motor, and for providing the resulting measurements as        a drive current intensity signal to the drive controller,        wherein the drive controller includes:    -   a pulse-width modulation generator;    -   a control law module storing the given control law; and    -   a state variable estimation module for estimating the        instantaneous value of at least one state variable of the        controlled AC electric motor,        wherein the control law module is adapted to, based on the        stored control law and state variable estimates provided by the        estimation module, compute a target voltage signal and output        the computed target voltage signal to the pulse-width modulation        generator,        wherein the pulse-width modulation generator is adapted to:    -   approximate the received target voltage signal with a        pulse-width modulated inverter control signal;    -   control the operation of the power inverter using the inverter        control signal, thereby obtaining the drive voltage;    -   compute, based on the deviation between the inverter control        signal and the target voltage signal, a state variable        estimation support signal; and    -   output the computed state variable estimation support signal to        the state variable estimation module, and        wherein the state variable estimation module is adapted to:    -   estimate the instantaneous value of a state variable of the AC        electric motor based on the received state variable estimation        support signal and the drive current intensity signal provided        by the drive current sensing device; and    -   output the resulting state variable estimate to the control law        module.

By modifying the PWM generator so that it computes an estimation supportsignal based on the PWM, and by providing this estimation support signalto the state variable estimator, the state variable estimator hassupplementary information, which it can use to improve its statevariable estimations.

Optionally, the variable speed drive according to the present disclosurecan have the following features, separately or in combination one withthe others:

-   -   the pulse-width modulation generator is adapted to compute the        state variable estimation support signal based on a pulse-width        modulation inherent disturbance signal, which is obtained by        subtracting the target voltage signal from the inverter control        signal;    -   the pulse-width modulation generator is adapted to compute the        state variable estimation support signal by integrating the        disturbance signal to obtain the primitive of the disturbance        signal;    -   the variable speed drive is adapted to rely on a single feedback        to perform closed loop control of the AC electric motor, namely        the drive current intensity signal provided by the drive current        sensing device;    -   the variable speed drive is adapted to control the operation of        the AC electric motor without the injection of a dedicated        probing signal into the drive voltage;    -   the drive controller further includes an analog-to-digital        converter for converting the drive current intensity signal into        a digital signal prior to its input into the state variable        estimation module;    -   the state variable estimation module is adapted to estimate the        instantaneous value of the rotor position of the electric motor        based on the received state variable estimation support signal        and the drive current intensity signal provided by the drive        current sensing device;    -   the pulse-width modulation generator is adapted to apply        three-phase pulse-width modulation with single carrier to        generate the inverter control signal;    -   the pulse-width modulation generator is adapted to apply        three-phase pulse-width modulation with interleaved carriers to        generate the inverter control signal.

According to a further aspect, the present disclosure also relates to anelectric drive assembly comprising a synchronous reluctance motor or apermanent-magnet synchronous motor and a variable speed drive as definedabove for controlling the motor.

According to yet a further aspect, the present disclosure also relatesto a method of controlling, in a closed loop, the operation of an ACelectric motor based on a given control law, the method comprising thefollowing steps:

-   a) measuring the instantaneous intensity of the drive current taken    up by the controlled AC electric motor;-   b) estimating the instantaneous value of a state variable of the AC    electric motor using the measured drive current intensity;-   c) computing, based on the given control law and the estimated state    variable, a target voltage signal;-   d) approximating the computed target voltage signal with a    pulse-width modulated inverting control signal;-   e) computing, based on the deviation between the inverting control    signal and the target voltage signal, a state variable estimation    support signal;-   f) generating, by voltage inversion, a controlled alternating drive    voltage using the inverting control signal; and-   g) delivering the generated drive voltage to the controlled AC    electric motor;    wherein the state variable estimation according to step b) relies on    the state variable estimation support signal computed in step e) as    an additional input together with the drive current intensity    measured in step a).

BRIEF DESCRIPTION OF DRAWINGS

Other features, details and advantages will be shown in the followingdetailed description and in the figures, in which:

FIG. 1 is a block diagram of an electric drive assembly of the presentdisclosure, with a variable speed drive and an AC electric motor.

FIG. 2 is a block diagram illustrating the signal flow and processing inthe electric drive assembly of FIG. 1, and FIGS. 3 through 22 illustrateexample graphs.

DESCRIPTION OF EMBODIMENTS I. Exemplary Setup of a Variable Speed DriveAccording to the Present Disclosure

FIG. 1 is a schematic diagram of an electric drive assembly 100according to the present disclosure. The electric drive assembly 100comprises a variable speed drive, or VSD, 200 and an AC electric motor300.

The electric drive assembly 100 may be used in diverse industrialsettings. For example, it may drive a fan of a heating, ventilation andair conditioning (HVAC) system. As another example, it may also be usedto drive a water pump of a sewage installation. Many other industrialapplications can be envisaged by the skilled person.

Preferably, the AC electric motor 300 is a synchronous motor, such as apermanent magnet synchronous motor, or PMSM, or a synchronous reluctancemotor, or SynRM.

The purpose of the variable speed drive 200 is to control the properoperation of the electric motor 300. Thanks to the variable speed drive200, the motor 300 can be operated at the right speed at the right time,depending on the application. The variable speed drive 200 may alsoallow to control the torque output of the electric motor 300 to itsload.

The variable speed drive 200 controls the electric motor 300 in a closedloop. This means that the variable speed drive 200 constantly receivesfeedback on the instantaneous status of the motor 300 during the controlof the motor. The variable speed drive 200 adjusts its control of theelectric motor 300 based on a given control law. The specifics of thecontrol law depend on the type of application of the electric motor 300.

With reference to FIG. 1, the variable speed drive 200 comprises anoutput terminal 210, a solid-state power inverter 220, a drivecontroller 230, and a drive current sensing device or current sensor240.

The variable speed drive 200 is electrically connected to the electricmotor 300 via its output terminal 210. The power output 210 delivers acontrolled alternating drive voltage u_(pwm) to the AC electric motor300. The drive voltage u_(pwm) is a modulated signal whose amplitude isdetermined by the DC voltage Vbus applied to the power inverter 220. Themodulation frequency of the drive voltage u_(pwm) depends on theswitching frequency of the power inverter 220. The modulated drivevoltage u_(pwm) emulates an ideal sinusoidal drive voltage whoseamplitude and frequency determine the operation of the electric motor300.

The power inverter 220 generates the drive voltage u_(pwm) by choppingup a DC voltage with the help of solid-state switches T1, T2.

The skilled person will note that the diagram of FIG. 1 shows a singlephase control. This is only for simplification. Typically, the electricmotor 300 will be a three-phase motor. In this case, the power inverter220 generates a drive voltage for each of the three phases of the motor.

The current sensor 240 of the VSD 200 measures the instantaneousintensity of the drive current taken up by the electric motor 300. Thecurrent sensor 240 provides its measurements as a drive currentintensity signal i_(s) to the drive controller 230.

According to the present disclosure, the motor control by the VSD 200 isa so-called “sensorless” control. This means that the control feedbackentirely relies on the current measurements provided by the currentsensor 240. There are no external sensors mounted on the motor 300, suchas shaft encoders and the like, to provide feedback to the VSD 200 onthe motor status.

The drive controller 230 controls the generation of the drive voltageu_(pwm) by the power inverter 220. This is done on the basis of aninverter control signal M provided by the drive controller 230 to thepower inverter 220.

The drive controller 230 may be implemented as a microcontroller or afield programmable gate array (FPGA).

According to the present disclosure, the drive controller 230 includes apulse-width modulation, or PWM, generator 232, a control law module 234storing the given control law, a state variable estimation module 236,and an analog to digital converter, or ADC, 238.

The control law module 234 is adapted to, based on the stored controllaw and state variable estimates z₀ to z_(n) provided by the estimationmodule 236, compute a target voltage signal u_(s) and output thecomputed target voltage signal u_(s) to the PWM generator 232.

The target voltage signal u_(s) represents the analog voltages that mustbe applied to the stator windings of the electric motor 300 to obtainthe desired speed or torque from the motor 300.

Since the variable speed drive 200 relies on pulse-width modulation, thetarget voltage signal u_(s) is not directly applied to the electricmotor 300. Rather, it is fed to the PWM generator 232 to be approximatedby the pulse-width modulated inverter control signal M, which in turn isused to control the power inverter 220.

The pulse-width modulation generator 232 may apply three-phasepulse-width modulation with single carrier to generate the invertercontrol signal M (i.e. the approximation of the target voltage signalu_(s)).

Alternatively, the PWM generator may also apply three-phase pulse-widthmodulation with interleaved carriers to generate the inverter controlsignal M.

The PWM generator 232 may of course also use other PWM schemes togenerate the inverter control signal M.

According to the present disclosure, the PWM generator 232 has theparticularity that it computes, based on the deviation between theinverter control signal M and the target voltage signal u_(s), a statevariable estimation support signal s₁, and outputs the computed statevariable estimation support signal s₁ to the state variable estimationmodule 236.

The state variable estimation module or estimator 236 estimates theinstantaneous value of one or more state variables of the AC electricmotor 300 based on the drive current intensity signal i_(s) provided bythe drive current sensor 240.

As shown in FIGS. 1 and 2, the estimator 236 may estimate several statevariables z₀ to z_(n). These state variables may e.g. correspond to therotor position of the electric motor, the angular velocity of themotor's rotor, etc.

According to the present disclosure, the estimator 236 also uses theestimation support signal s₁ to estimate the value of at least one ofthe state variables z₀ to z_(n).

The estimator 236 provides the state variable estimates z₀ to z_(n) tothe control law module 234. The control law module 234 uses theseestimates in the stored control law in order to determine the targetvoltage signal u_(s).

As shown in the figures, the drive controller 230 may also include ananalog to digital converter 238. The purpose of the ADC 238 is toconvert the analog current signal i_(s) provided by the current sensor240 into a digital signal that can be processed by the estimator 236.

FIG. 2 shows the signal flow between the different components of theelectric drive assembly 100 of FIG. 1. The PWM generator 232 receivesthe target voltage signal u_(s) from the control law module 234. Usingpulse-width modulation, it approximates the target voltage signal u_(s)by an inverter control signal M (u, t/ε). The inverter control signal Mis fed to the power inverter 220. Based on this control signal M, theinverter 220 delivers a drive voltage u_(pwm) to the electric motor 300.With the current sensor 240, the currents in the motor's stator windingsare measured and digitized in the ADC 238. The digitized current signalis then fed to the estimator 236. The estimator 236 also receives theestimation support signal s₁ (u, t/ε) from the PWM generator 232. Theestimator 236 provides estimates of different motor state variables z₀to z_(n) based on the received inputs.

An important aspect of the present disclosure is the enhanced PWMgenerator 232, which not only generates the inverter control signal M,but also the state variable estimation support signal s₁. With the helpof the estimation support signal s₁, the estimation module 236 canextract supplementary information from the current signal i_(s) toimprove the state variable estimations.

The PWM generator 232 computes the state variable estimation supportsignal s₁ based on a pulse-width modulation inherent disturbance signals₀. In fact, the present disclosure relies on the insight that theinverter control signal M generated by the PWM generator 232 can bemodelled as a superposition of the target voltage signal u_(s) and adisturbance signal s₀. Indeed, the inverter control signal M is a seriesof rectangular voltage pulses of varying widths, which on average,corresponds to the desired target voltage signal u_(s). In other words,the inverter control signal M can be regarded as the desired targetvoltage signal u_(s) with an added voltage “ripple”. This ripple or PWMdisturbance in the voltage creates a disturbance in the stator flux ofthe motor 300, which in turn creates a disturbance in the measuredcurrent i_(s).

The present disclosure takes advantage of this unintended side-effect ofthe pulse width modulation. The ripple/artefact induced by thepulse-width modulation in the measured currents is used in the statevariable estimation. This improves the estimates and thus the control ofthe electric motor 300.

A measure of the pulse-width modulation disturbance signal s₀ (i.e. theripple) can be obtained by subtracting the target voltage signal u_(s)from the inverter control signal M. The difference between the targetvoltage signal us and the inverter control signal M can then beintegrated to obtain the primitive s₁ of the disturbance signal s₀.

These calculations, namely the subtraction and the integration, are doneby the PWM generator 232. As shown in the figures, the PWM generator 232provides the result s₁ to the estimator 236.

It can be mathematically shown (see the remainder of the presentdisclosure) that the primitive s₁ is a useful input for the estimator236 to determine the instantaneous value of a state variable of theelectric motor 300, such as the rotor position θ.

The variable speed drive of the present disclosure is particularlyuseful for the control of synchronous electric motors at low speeds. Bysuitably using the excitation provided by the pulse-width modulationitself, the variable speed drive of the present disclosure has the samebenefits as a conventional variable speed drive relying on an externalexcitation signal, without the drawbacks of increased acoustic noise andpotential interference with the pulse width modulation.

In the variable speed drive of the present disclosure, standardpulse-width modulation does not need to be modified.

Also, the estimates by the estimation module 236 only require currentmeasurements from standard current sensors. There is no need for currentmeasurements at extremely precise instants, which is prone tomeasurement errors and very impractical in an industrial drive.

Furthermore, the variable speed drive of the present disclosure alsodoes not require specialized sensors capable of measuring currentderivatives, as in other known solutions.

The teachings of the present disclosure may also be applied to thecontrol of other types of actuators. For example, one may think ofcontrolling the operation of an electromagnet in a magnetic bearing, orof a solenoid valve of a hydraulic or pneumatic cylinder.

II. Mathematical Derivation of Exemplary Estimators for SensorlessElectric Motor Control

-   Consider the control system modeled by the evolution equation    ż=f(z)+g(z)u    y _(α) =h(z)  (1)    where z is the internal state vector of the system, u is the control    input vector, and y_(α) is the measurement vector. In several    practical applications (in particular electromechanical devices fed    by PWM inverter), the control input is not applied directly, but    through a fast-periodic modulation which yields u only in the mean.    The actual system is therefore    {dot over (x)}=f(x)+g(x)    (u,t/ϵ)    y=h(x)    where x is the state vector of the system disturbed by the modulator    and y the corresponding measurement. ϵ>0 is a known small number,    and    is 1-periodic and has mean u with respect to its second argument,    i.e.,    (ν, σ)=    (ν, σ+1) and ∫₀ ¹    (ν, σ)dσ=ν. The system can clearly be written as    {dot over (x)}=f(x)+g(x)(u+s ₀(u,t/ϵ))    y=h(x)    where s₀ defined by s₀(ν, σ):=    (ν,σ)−ν is 1-periodic and has zero mean with respect to its second    argument, i.e., s₀(ν, σ)=s₀(ν, σ+1) and ∫₀ ¹s₀(ν, σ)dσ=0. In other    words, we consider a generalized kind of signal injection, where the    probing signal s₀ is generated by the modulation process itself;    notice s₀ depends on u, which causes several difficulties.-   Thanks to a mathematical analysis based on the cherry of    second-order averaging, it can be shown that the measured signal y    may be written as    y(t)=y _(α)(t)+ϵy _(ν() t)s ₁(u(t),t/ϵ)+O(ϵ²)    where defined by s₁, defined by s₁(ν, σ):=∫₀ ^(σ)s₀(ν, s)ds−∫₀ ¹∫₀    ^(σ)s₀(ν, s)ds dσ, is the zero-mean primitive of s₀ with u    considered as a parameter; the “actual measurement” y_(α) and the    “virtual measurement” y_(ν) s₁s₁ ^(T) can be interpreted as outputs    of the “ideal” system without modulation (1), namely    y _(α) =h(z)  (2)    y _(ν) s ₁ s ₁ ^(T) (u)=h′(z)g(z) s ₁ s ₁ ^(T) (u)  (3)    Notice the square matrix s₁s₁ ^(T) (u)=∫₀ ¹(s₁s₁ ^(T))(u, σ)dσ is in    general invertible (see “Multiphase PWM” hereafter), so that we    simply have y_(ν)=h′(z)g(z).-   The overall purpose of the invention is to extract from the measured    signal y estimates of y_(α) and y_(ν) which can be used to control    the system. All is then as if one would want to control the “ideal”    system without modulation (1) with a feedback law relying not only    on the “actual measurement” (2), but also on the “virtual    measurement” (3). The problem is then simpler thanks to the    supplementary information (3); in particular in the case of    sensorless control electric motors at low velocity, this    supplementary information gives access to the rotor position, hence    is instrumental in the design of the control law. The invention    consists mainly of two parts:    -   1) an enhanced modulator, generating not only the modulating        signal        , but also the zero-mean primitive s₁ of s₀    -   2) estimators        and        extracting not only y_(α), but also y_(ν) from the measured        signal y, using the knowledge of s1.-   A possible implementation for the estimators is

$\begin{matrix}{{(t)} = {{\frac{3}{2\epsilon}{\int_{t - \epsilon}^{t}{{y(\tau)}d\;\tau}}} - {\frac{1}{2\epsilon}{\int_{t - {2\epsilon}}^{t - \epsilon}{{y(\tau)}{dt}}}}}} \\{{(t)\overset{—––}{s_{1}s_{1}^{T}}\left( {u(t)} \right)} = {\frac{1}{\epsilon^{2}}{\int_{t - \epsilon}^{t}{\left( {{y(\tau)} - {(\tau)}} \right){s_{1}\left( {{u(\tau)},{\tau/\epsilon}} \right)}d\;\tau}}}}\end{matrix}$related to the desired signals by

(t)=y _(α)(t)+

(ϵ²)

(t) s ₁ s ₁ ^(T) (u(t))=y _(ν)(t) s ₁ s ₁ ^(T) (u(t))+

(ϵ).Notice these estimators are periodic low-pass filters with FIR (FiniteImpulse response) many variants relying on different periodic low-passFIR filters are possible.

III. Pulse-Width Modulation (PWM)

Generating analog physical power signals is extremely impractical, sincea lot of power must be dissipated in the amplifier. Pulse WidthModulation (PWM) addresses the issue by using the transistors insaturation mode. Indeed, transistors are more efficient when they areused in saturation than when they are used in their linear range. Thedesired value is realized in average by adjusting the width of thepulses (hence the name of the technique).

The simplest way to realize a PWM modulator is to compare the analogsignal u with a triangular carrier c oscillating between −u_(max) andu_(max) follows:

$\begin{matrix}{{u_{PWM}(t)} = \left\{ {\begin{matrix}{{{- u_{\max}}\mspace{14mu}{if}\mspace{14mu}{c(t)}} > {u(t)}} \\{{u_{\max}\mspace{14mu}{if}\mspace{14mu}{c(t)}} < {u(t)}}\end{matrix} = {u_{\max{\;\mspace{11mu}}}{sign}\mspace{14mu}\left( {{u(t)} - {c(t)}} \right)}} \right.} \\{where} \\{{c(t)} = \left\{ \begin{matrix}{{4{u_{\max}\left( {\frac{t}{T} - k + \frac{1}{4}} \right)}\mspace{14mu}{if}\mspace{14mu}\frac{t}{T}} \in \left\lbrack {{k - \frac{1}{2}},k} \right\rbrack} \\{{{- 4}{u_{\max}\left( {\frac{t}{T} - k - \frac{1}{4}} \right)}\mspace{14mu}{if}\mspace{14mu}\frac{t}{T}} \in \left\lbrack {k,{k + \frac{1}{2}}} \right\rbrack}\end{matrix} \right.}\end{matrix}$

This is the natural sampling PWM demonstrated in FIG. 10 where thecarrier is shown as continuous line c, the analog signal is shown asdashed line u and the PWM signal is shown as dotted line Upwm.

One of the disadvantages of the natural sampling PWM is that the pulsesare not symmetrical. To address this issue, we can first sample theanalog signal to obtain u[k]=u(kT) and then apply the PWM modulation onthe sampled signal. With this PWM scheme we can check that

$\begin{matrix}{{\frac{1}{T}{\int_{{({k - \frac{1}{2}})}T}^{{({k + \frac{1}{2}})}T}{{u_{PWM}(t)}{dt}}}} = {\frac{1}{T}\left( {{\int_{{({k - \frac{1}{2}})}T}^{{({\frac{u{\lbrack k\rbrack}}{4u_{\max}} + k - \frac{1}{4}})}T}{u_{\max}{dt}}} -} \right.}} \\{{\int_{{({\frac{u{\lbrack k\rbrack}}{4u_{\max}} + k - \frac{1}{4}})}T}^{{({{- \frac{({u{\lbrack k\rbrack}}}{4u_{\max}}} + k + \frac{1}{4}})}T}{u_{{ma}x}{dt}}} +} \\\left. {\int_{{({{- \frac{u{\lbrack k\rbrack}}{4u_{\max}}} + k + \frac{1}{4}})}T}^{{({k + \frac{1}{2}})}T}{u_{\max}{dt}}} \right) \\{= {\frac{u_{\max}}{T}\left( {\frac{u\lbrack k\rbrack}{4u_{\max}} + \frac{1}{4} + {2\frac{u\lbrack k\rbrack}{4u_{\max}}} - \frac{1}{2} +} \right.}} \\\left. {\frac{u\lbrack k\rbrack}{4u_{\max}} + \frac{1}{4}} \right) \\{= {u\lbrack k\rbrack}}\end{matrix}$

The expression of u_(PWM) can be rewritten under the form u_(PWM)=u+ũwith

${{\overset{\sim}{u}\left( {\frac{t}{T},u} \right)} = {{u_{\max}\mspace{14mu}{{sign}\left( {u - {c\left( \frac{t}{T} \right)}} \right)}} - u}},$which is a zero-mean fast-varying periodic signal with a dependence onthe control.

IV. Poly-Phase PWM

When controlling poly-phase electrical devices, n>1 analog referencesmust be modulated.

In this case the s₁ will be a vector with n lines and s₁s₁ ^(T) is a n×nmatrix.

Traditionally, to facilitate implementation, the n modulators use thesame carrier. In this case, when two of the references are equal, thiscontrol scheme leads to two equal components in s₁, which means thats₁s₁ ^(T) , will not be invertible, which means that we do not obtainall the information in h′(z)g(z). Besides, when two components are closefrom one another s₁s₁ ^(T) will be poorly conditioned, whichcomplexities the signal processing.

To get more information and improve the conditioning of s₁s₁ ^(T) , wecan use a different carrier for each reference. In this case, thecomponents of s₁ are always independent. Consequently, the matrix s₁s₁^(T) , is always invertible and well-conditioned.

V. Application to Sensorless Control of an Unsaturated SynchronousReluctance Motor (SynRM)

Using Clarke transformation Matrices

${\mathcal{C}:={\frac{2}{3}\begin{pmatrix}1 & {{- 1}/2} & {{- 1}/2} \\0 & {\sqrt{3}/2} & {{- \sqrt{3}}/2}\end{pmatrix}}},{\mathcal{C}^{- 1} = \begin{pmatrix}1 & 0 \\{{- 1}/2} & {\sqrt{3}/2} \\{{- 1}/2} & {{- \sqrt{3}}/2}\end{pmatrix}}$and the rotation

${\mathcal{R}(\theta)}:=\begin{pmatrix}{\cos\mspace{14mu}\theta} & {{- \sin}\mspace{14mu}\theta} \\{\sin\mspace{14mu}\theta} & {\cos\mspace{14mu}\theta}\end{pmatrix}$the model of the Synchronous Reluctant Motor (SynRM) is given by

$\begin{matrix}{\frac{d\;\phi_{SDQ}}{dt} = {{{\mathcal{R}\left( {- \theta} \right)}\mathcal{C}\; u_{Sabc}} - {R_{s}L^{- 1}\phi_{SDQ}} - {J\;{\omega\phi}_{SDQ}}}} \\{\frac{d\;\theta}{dt} = \omega} \\{t_{Sabc} = {\mathcal{C}^{- 1}{\mathcal{R}(\theta)}L^{- 1}\phi_{SDQ}}}\end{matrix}$

The state of this system is ϕ_(SDQ), the vector of the stator flux infield-oriented DQ frame, and θ, the angular position of the rotor. Thevector of stator voltages in physical abc frame, u_(Sabc) is the controlinput, while ω, the rotor speed, is a disturbance input, which must beobtained to achieve a proper control of the SynRM. When “sensorless”control is used the sole available measurement is the vector of statorcurrents in physical abc frame, t_(Sabc). The parameters of the modelare the stator resistance R_(s) and the matrix of inductances

$L = {\begin{pmatrix}L_{D} & 0 \\0 & L_{Q}\end{pmatrix}.}$

Since PWM is used, the voltage can be writtenu_(Sabc)(t)=ū_(Sabc)(t)+ũ_(Sabc)(t/ϵ, ū_(Sabc)(t)).

Rewriting the system in the time scale σ:=t/ϵ, we obtain

$\begin{matrix}{\frac{d\;\phi_{SDQ}}{d\;\sigma} = {\epsilon\left( {{{\mathcal{R}\left( {- \theta} \right)}\mathcal{C}{\overset{\_}{u}}_{Sabc}} + {{\mathcal{R}\left( {- \theta} \right)}\mathcal{C}{{\overset{\_}{u}}_{Sabc}\left( {\sigma,{\overset{\_}{u}}_{Sabc}} \right)}} - {R_{s}L^{- 1}\phi_{SDQ}} - {J\;{\omega\phi}_{SDQ}}} \right)}} \\{\mspace{79mu}{\frac{d\;\theta}{d\;\sigma} = {\epsilon\omega}}} \\{\mspace{79mu}{t_{Sabc} = {\mathcal{C}^{- 1}{\mathcal{R}(\theta)}L^{- 1}\phi_{SDQ}}}}\end{matrix}$

Which is like the standard form of averaging. Applying the averagingprocedure, we can show that the PWM disturbance in the voltages createsa disturbance in the stator flux, which becomes ϕ_(SDQ)=ϕ _(SDQ)+ϵ

(−θ)

Ũ_(Sabc)(t/ϵ,ū_(Sabc)), where Ũ_(Sabc) is the primitive of ũ_(Sabc),with zero-mean over a PWM method. The flux disturbance in turn creates adisturbance in the measured currents which becomes

$t_{Sabc} = {{\overset{\_}{t}}_{Sabc} + {{\epsilon\mathcal{C}}^{- 1}\underset{s{(\theta)}}{\underset{︸}{\mathcal{R}(\theta)L^{- 1}{\mathcal{R}\left( {- \theta} \right)}}}\mathcal{C}{{\overset{\sim}{U}}_{Sabc}\left( {{t/\epsilon},{\overset{\_}{u}}_{Sabc}} \right)}} + {O\left( \epsilon^{2} \right)}}$

The undisturbed variables follow the original model

$\begin{matrix}{\frac{d{\overset{\_}{\phi}}_{SDQ}}{dt} = {{{\mathcal{R}\left( {- \theta} \right)}\mathcal{C}{\overset{\_}{u}}_{Sa\beta}} - {R_{s}L^{- 1}{\overset{\_}{\phi}}_{SDQ}} - {J\;\omega{\overset{\_}{\phi}}_{SDQ}}}} \\\begin{matrix}{\frac{d\;\theta}{d\; t} = \omega} \\{{\overset{\_}{t}}_{S\;{\alpha\beta}} = {\mathcal{C}^{- 1}{\mathcal{R}(\theta)}L^{- 1}{\overset{\_}{\phi}}_{SDQ}}}\end{matrix}\end{matrix}$

Thanks to the proposed estimators, provided Ũ_(Sabc)Ũ_(Sabc) ^(T) is theinvertible (see “Poly-phase PWM” above), we can recover t _(Sabc) and

(θ) from t_(Sabc), which allow to run the or control law, which computesū_(Sαβ) from t _(Sαβ), and retrieve θ using

(θ)=

(θ)L⁻¹

(−θ), as can be done with HF injection, but without the need of anadditional disturbance.

Adding Virtual Measurements by PWM-Induced Signal Injection

Abstract: We show that for PWM-operated devices, it is possible tobenefit from signal injection without an external probing signal, bysuitably using the excitation provided by the PWM itself. As in theusual signal injection framework conceptualized in [1], an extra“virtual measurement” can be made available for use in a control law,but without the practical drawbacks caused by an external signal.

I. Introduction

Signal injection is a control technique which consists in adding afast-varying probing signal to the control input. This excitationcreates a small ripple in the measurements, which contains usefulinformation if properly decoded. The idea was introduced in [2], [3] forcontrolling electric motors at low velocity using only measurements ofcurrents. It was later conceptualized in [1] as a way of producing“virtual measurements” that can be used to control the system, inparticular to overcome observability degeneracies. Signal injection is aeffective method, see e.g. applications to electromechanical devicesalong these lines in [4], [5], but it comes at a price; the ripple itcreates may in practice yield unpleasant acoustic noise and exciteunmodeled dynamics, in particular in the very common situation when thedevice is fed by a Pulse Width Modulation (PWM) inverter; indeed, thefrequency of the probing signal may not be as high as desired so as notto interfere with the PWM (typically, it can not exceed 500 Hz in anindustrial drive with a 4 kHz-PWM frequency).

The goal of this paper is to demonstrate that for PWM-operated devices,it is possible to benefit from signal injection without an externalprobing signal, by using the excitation provided by the PWM itself, ase.g. in [6]. More precisely, consider the Single-Input Single-Outputsystem{dot over (x)}=f(x)+g(x)u,  (1a)y=h(x),  (1b)where u is the control input and y the measured output. We first show insection II that when the control is impressed through PWM, the dynamicsmay be written as

$\begin{matrix}{{\overset{.}{x} = {{f(x)} + {{g(x)}\left( {u + {s_{0}\left( {u,\frac{t}{ɛ}} \right)}} \right)}}},} & (2)\end{matrix}$with s₀ 1-periodic and zero-mean in the second argument i.e. s₀(u,σ+1)=s₀(u, σ) and ∫₀ ¹s₀(u, σ)dσ=0 for all u; ε is the PWM period, henceassumed small. The difference with usual signal injection is that theprobing signal s₀ generated by the modulation process now depends notonly on time, but also on the control input u. This makes the situationmore complicated, in particular because s₀ can be discontinuous in bothits arguments. Nevertheless, we show in section III that thesecond-order averaging analysis of [1] can be extended to this case. Inthe same way, we show in section IV that the demodulation procedure of[1] can be adapted to make available the so-called virtual measurementy ₀ :=H ₁(x):=εh′(x)g(x),in addition to the actual measurement y₀:=H₀(x):=h(x). This extra signalis likely to simplify the design of a control law, as illustrated on anumerical example in section V.

Finally, we list some definitions used throughout the paper, S denotes afunction of two variables, which is T-periodic in the second argument,i.e. S(ν, σ+T)=S(ν, σ) for all ν:

-   -   the mean of S in the second argument is the function (of one        variable)

${{\overset{\_}{S}(v)}:={\frac{1}{T}{\int_{0}^{T}{{S\left( {v,\sigma} \right)}d\;\sigma}}}};$S has zero mean in the second argument if S is identically zero

-   -   if S has zero mean in the second argument, its zero-mean        primitive in the second argument is defined by

${{S_{1}\left( {v,\tau} \right)}:={{\int_{0}^{\tau}{{S\left( {v,\sigma} \right)}d\;\sigma}} - {\frac{1}{T}{\int_{0}^{T}{\int_{0}^{\tau}{{S\left( {v,\sigma} \right)}d\;\sigma\; d\;\tau}}}}}};$notice S₁ is T-periodic in the second argument because S has zero meanin the second argument

-   -   the moving average M(k) of k is defined by

${{M(k)}(t)}:={\frac{1}{ɛ}{\int_{t - ɛ}^{t}{{k(\tau)}d\;\tau}}}$

-   -   _(∞) denotes the uniform “big O” symbol of analysis, namely ƒ(z,        ε)=        _(∞)(ε^(p)) if |ƒ(z, ε)|≤Kε^(p) for ε small enough, with K>0        independent of z and ε.

II. PWM-Induced Signal Injection

When the control input u in (1a) is impressed through a PWM process withperiod ε, the resulting dynamics reads

$\begin{matrix}{{\overset{.}{x} = {{f(x)} + {{g(x)}{\mathcal{M}\left( {u,\frac{t}{ɛ}} \right)}}}},} & (3)\end{matrix}$with

1-periodic and mean u in the second argument; the detailed expressionfor

is given below. Setting s₀(u, σ):=

(u, σ)−u, (3) obviously takes the form (2), with s₀ 1-periodic andzero-mean in the second argument.

Classical PWM with the period ε and range [−u_(m), u_(m)] is obtained bycomparing the input signal u to the ε-periodic sawtooth carrier definedby

${c(t)}:=\left\{ {\begin{matrix}{u_{m} + {4{w\left( \frac{t}{ɛ} \right)}}} & {{{if}\mspace{14mu} - \frac{u_{m}}{2}} \leq {w\left( \frac{t}{ɛ} \right)} \leq 0} \\{u_{m} - {4{w\left( \frac{t}{ɛ} \right)}}} & {{{if}\mspace{14mu} 0} \leq {w\left( \frac{t}{ɛ} \right)} \leq \frac{u_{m}}{2}}\end{matrix};} \right.$the 1-periodic function

${w(\sigma)}:={{u_{m}{{mod}\left( {{\sigma + \frac{1}{2}},1} \right)}} - \frac{u_{m}}{2}}$wraps the normalized time

$\sigma = {\frac{t}{ɛ}\mspace{14mu}{{{to}\mspace{14mu}\left\lbrack {{- \frac{u_{m}}{2}},\frac{u_{m}}{2}} \right\rbrack}.}}$If u varies slowly enough, it crosses the carrier c exactly once on eachrising and filling ramp, at times t₁ ^(u)>t₂ ^(u) such that

${u\left( t_{1}^{u} \right)} = {u_{m} + {4{w\left( \frac{t_{1}^{u}}{ɛ} \right)}}}$${u\left( t_{2}^{u} \right)} = {u_{m} - {4{{w\left( \frac{t_{2}^{u}}{ɛ} \right)}.}}}$

The PWM-encoded signal is therefore given by

${u_{pwm}(t)} = \left\{ {\begin{matrix}u_{m} & {{{if}\mspace{14mu} - \frac{u_{m}}{2}} < {w\left( \frac{t}{ɛ} \right)} \leq {w\left( \frac{t_{1}^{u}}{ɛ} \right)}} \\{- u_{m}} & {{{if}\mspace{14mu}{w\left( \frac{t_{1}^{u}}{ɛ} \right)}} < {w\left( \frac{t}{ɛ} \right)} \leq {w\left( \frac{t_{2}^{u}}{ɛ} \right)}} \\u_{m} & {{{if}\mspace{14mu}{w\left( \frac{t_{2}^{u}}{ɛ} \right)}} < {w\left( \frac{t}{ɛ} \right)} \leq \frac{u_{m}}{2}}\end{matrix}.} \right.$FIG. 3 illustrates the signals u, c and u_(pwm). The function

$\begin{matrix}{{\mathcal{M}\left( {u,\sigma} \right)}:=\left\{ \begin{matrix}u_{m} & {{{if}\mspace{14mu} - {2u_{m}}} < {4{w(\sigma)}} \leq {u - u_{m}}} \\{- u_{m}} & {{{{if}\mspace{14mu} u} - u_{m}} < {4{w(\sigma)}} \leq {u_{m} - u}} \\u_{m} & {{{{if}\mspace{14mu} u_{m}} - u} < {4{w(\sigma)}} \leq {2u_{m}}}\end{matrix} \right.} \\{{= {u_{m} + {{sign}\left( {u - u_{m} - {4{w(\sigma)}}} \right)} + {{sign}\left( {u - u_{m} + {4{w(\sigma)}}} \right)}}},}\end{matrix}$which is obviously 1-periodic and with mean u with respect to its secondargument, therefore completely describes the PWM process since

${u_{pwm}(t)} = {{\mathcal{M}\left( {{u(t)},\frac{t}{ɛ}} \right)}.}$

Finally, the induced zero-mean probing signal is

$\begin{matrix}{{s_{0}\left( {u,\sigma} \right)}:={{\mathcal{M}\left( {u,\sigma} \right)} - u}} \\{{= {u_{m} - u + {{sign}\left( {\frac{u - u_{m}}{4} - {w(\sigma)}} \right)} + {{sign}\left( {\frac{u - u_{m}}{4} - {w(\sigma)}} \right)}}},}\end{matrix}$and its zero-mean primitive in the second argument is

${s_{1}\left( {u,\sigma} \right)}:={{\left( {1 - \frac{u}{u_{m}}} \right){w(\sigma)}} - {{\frac{u - u_{m}}{4} - {w(\sigma)}}} + {{{\frac{u - u_{m}}{4} - {w(\sigma)}}}.}}$

Remark 1: As s₀ is only piecewise continuous, one might expect problemsto define the “solutions” of (2). But as noted above, if the input u(t)of the PWM encoder varies slowly enough, its output

${u_{pwm}(t)} = {\mathcal{M}\left( {{u(t)},\frac{t}{ɛ}} \right)}$will have exactly two discontinuities per PWM period. Chattering istherefore excluded, which is enough to ensure the existence anduniqueness of the solutions of (2), see [7], without the need for themore general Filipov theory [8]. Of course, we assume (without loss ofgenerality in practice) that ƒ, g and h in (1) are smooth enough.

Notice also s₁ is continuous and piecewise C¹ in both its arguments. Theregularity in the second argument was to be expected as s₁(u, ·) is aprimitive of s₀(u, ·); on the other hand, the regularity in the firstargument stems from the specific form of s₀.

III. Averaging and PWM-Induced Injection

Section III-A outlines the overall approach and states the main Theorem1, which is proved in the somewhat technical section III-B. As a matterof fact, the proof can be skipped without losing the main thread;suffice to say that if s₀ were Lipschitz in the first argument, theproof would essentially be an extension of the analysis by “standard”second-order averaging of [1], with more involved calculations

A. Main Result

Assume we have designed a suitable control lawū=α(η, Y,t){dot over (η)}=α(η, Y,t)₁where ηε

^(q), for the system

${\overset{.}{\overset{\_}{x}} = {{f\left( \overset{\_}{x} \right)} + {{g\left( \overset{\_}{x} \right)}\overset{\_}{u}}}},{\overset{\_}{Y} = {{H\left( \overset{\_}{x} \right)}:={\begin{pmatrix}{h\left( \overset{\_}{x} \right)} \\{ɛ\;{h^{\prime}\left( \overset{\_}{x} \right)}{g\left( \overset{\_}{x} \right)}}\end{pmatrix}.}}}$By “suitable”, we mean the resulting closed-loop system{dot over (x)} =ƒ( x )+g( x )α(η,H( x ),t)  (4a){dot over (η)}=α(η,H( x ),t)  (4b)has the desired exponentially stable behavior. We have changed thenotations of the variables with · to easily distinguish between thesolutions of (4) and of (7) below. Of course, this describes anunrealistic situation:

-   -   PWM is not taken into account    -   the control law is not implementable, as it uses not only the        actual output y _(α)=h(x), but also the a priori not available        virtual output y _(v)=εh′(x)g(x).

Define now (up to

_(∞)(ε²)) the functionH (x,η,σ,t):=H(x−εg(x)s ₁(α(η,H(x),t),σ))+

_(∞)(ε²),  (5)where s₁ is the zero mean primitive of s₀ in the second argument, andconsider the control law

$\begin{matrix}{u = {\alpha\left( {\eta,{\overset{\_}{H}\left( {x,\eta,\frac{t}{ɛ},t} \right)},t} \right)}} & \left( {6a} \right) \\{\overset{.}{\eta} = {{a\left( {\eta,{\overset{\_}{H}\left( {x,\eta,\frac{t}{ɛ},t} \right)},t} \right)}.}} & \left( {6b} \right)\end{matrix}$The resulting closed-loop system, including PWM, reads

$\begin{matrix}{\overset{.}{x} = {{f(x)} + {{g(x)}{\mathcal{M}\left( {{\alpha\left( {\eta,{\overset{\_}{H}\left( {x,\eta,\frac{t}{ɛ},t} \right)},t} \right)},\frac{t}{ɛ}} \right)}}}} & \left( {7a} \right) \\{\overset{.}{\eta} = {{a\left( {\eta,{\overset{\_}{H}\left( {x,\eta,\frac{t}{ɛ},t} \right)},t} \right)}.}} & \left( {7b} \right)\end{matrix}$Though PWM is now taken into account the control law (6) still seems tocontain unknown terms.Nevertheless, it will turn out from the following result that it can beimplemented.

Theorem 1: Let (x(t), η(t)) be the solution of (7) starting from (x₀,η₀), and define u(t):=α(η(t), H(x(t)), t) and y(t):=H(x(t)); let (x(t),η(t)) be the solution of (4) starting from (x₀−εg(x₀). s₁(u(0), 0), η₀),and define ū(t):=α(η(t), H(x(t)),t). Then, for t≥0,

$\begin{matrix}{{x(t)} = {{\overset{\_}{x}(t)} + {ɛ\;{g\left( {\overset{\_}{x}(t)} \right)}{s_{1}\left( {{\overset{\_}{u}(t)},\frac{t}{ɛ}} \right)}} + {\mathcal{O}_{\infty}\left( ɛ^{2} \right)}}} & \left( {8a} \right) \\{{\eta(t)} = {{\overset{\_}{\eta}(t)} + {\mathcal{O}_{\infty}\left( ɛ^{2} \right)}}} & \left( {8b} \right) \\{{y(t)} = {{H_{0}\left( {\overset{\_}{x}(t)} \right)} + {{H_{1}\left( {\overset{\_}{x}(t)} \right)}{s_{1}\left( {{\overset{\_}{u}(t)},\frac{t}{ɛ}} \right)}} + {{\mathcal{O}_{\infty}\left( ɛ^{2} \right)}.}}} & \left( {8c} \right)\end{matrix}$

The practical meaning of the theorem is the following. As the solution(x(t), η(t))) is piecewise C¹, we have by Taylor expansion using(8a)-(8b) that u(t)=ū(t)+

_(∞)(ε²). In the same way, as s₁ is also piecewise C¹, we have

${s_{1}\left( {{u(t)},\frac{t}{ɛ}} \right)} = {{s_{1}\left( {{\overset{\_}{u}(t)},\frac{t}{ɛ}} \right)} + {{\mathcal{O}_{\infty}\left( ɛ^{2} \right)}.}}$As a consequence, we can invert (8a)-(8b), which yields

$\begin{matrix}{{\overset{\_}{x}(t)} = {{x(t)} - {ɛ\;{g\left( {x(t)} \right)}{s_{1}\left( {{u(t)},\frac{t}{ɛ}} \right)}} + {\mathcal{O}_{\infty}\left( ɛ^{2} \right)}}} & \left( {9a} \right) \\{{\overset{\_}{\eta}(t)} = {{\eta(t)} + {{\mathcal{O}_{\infty}\left( ɛ^{2} \right)}.}}} & \left( {9b} \right)\end{matrix}$Using this into (5), we then get

$\begin{matrix}\begin{matrix}{{{\overset{\_}{H}\left( {{x(t)},{\eta(t)},\frac{t}{ɛ},t} \right)} = {{H\left( {{x(t)} - {ɛ\;{g\left( {x(t)} \right)}{s_{1}\left( {{u(t)},\frac{t}{ɛ}} \right)}}} \right)} + {\mathcal{O}_{\infty}\left( ɛ^{2} \right)}}},} \\{= {{H\left( {\overset{\_}{x}(t)} \right)} + {{\mathcal{O}_{\infty}\left( ɛ^{2} \right)}.}}}\end{matrix} & (10)\end{matrix}$On the other hand, we will see in section IV that, thanks to (8c), wecan produce an estimate Ŷ=H(x)+

_(∞)(ε²). This means the PWM-fed dynamics (3) acted upon by theimplementable feedbacku=α(η,Ŷ,t){dot over (η)}=α(η,Ŷ,t).behaves exactly as the closed-loop system (4) except for the presence ofa small ripple (described by (8a)-(8b)).

Remark 2: Notice that, according to Remark 1, H₀(x(t)) and H₁(x(t)) in(8c) may be as smooth as desired (the regularity is inherited from onlyf, g, h, α, a); on the other hand

$s_{1}\left( {{u(t)},\frac{t}{ɛ}} \right)$is only continuous and piecewise C¹. Nevertheless, this is enough tojustify all the Taylor expansions performed in the paper.Proof of Theorem 1

Because of the lack of regularity of s₀, we must go hack to thefundamentals of the second-order averaging theory presented in [9,chapter 2] (with slow time dependence [9, section 3.3]). We firstintroduce two ad hoc definitions.

Definition 1: A function φ(X, σ) is slowly-varying in average if thereexists λ>0 such that for ε small enough,∫_(α) ^(α+T)∥φ(p(εσ)+ε^(k) q(σ),σ)−φ(p(εσ),σ)∥dσ≤λTε ^(k),where p, q are continuous with q bounded; α and T>0 are arbitraryconstants. Notice that if φ is Lipschitz in the first variable then itis slowly-varying in average. The interest of this definition is that itis satisfied by s₀.

Definition 2: A function φ is

_(∞)(ε³) in average if there exists K>0 such that ∥∫₀ ^(α)ϕ(q(s),s)ds∥≤k ε³σ for all σ≥0. Clearly, if ϕ is

_(∞)(ε³) then it is

_(∞)(ε³) in average.

The proof of Theorem 1 follows the same steps as [9, chapter 2], butwith weaker assumptions. We first rewrite (7) in the fast timescaleσ:=t/ε as

$\begin{matrix}{{\frac{dX}{d\;\sigma} = {{ɛ\;{{F\left( {X,\sigma,{ɛ\sigma}} \right)}.{where}}\mspace{14mu} X}:={\left( {x,\eta} \right)\mspace{14mu}{and}}}}{{F\left( {X,\sigma,\tau} \right)}:={\begin{pmatrix}{{f(x)} + {{g(x)}{\mathcal{M}\left( {\alpha\left( {\eta,{\overset{\_}{H}\left( {x,\eta,\sigma,\tau} \right)},\tau} \right)} \right)}}} \\{a\left( {\eta,{\overset{\_}{H}\left( {x,\eta,\sigma,\tau} \right)},\tau} \right)}\end{pmatrix}.}}} & (11)\end{matrix}$Notice F is 1-periodic in the second argument. Consider also theso-called averaged system

$\begin{matrix}{\frac{d\overset{\_}{X}}{d\;\sigma} = {ɛ\;{{\overset{\_}{F}\left( {\overset{\_}{X},{ɛ\sigma}} \right)}.}}} & (12)\end{matrix}$where F is the mean F in the second argument.

Define the near-identity transformation

$\begin{matrix}{{{X = {\overset{\sim}{X} + {ɛ\;{W\left( {\overset{\sim}{X},\sigma,{ɛ\sigma}} \right)}}}},{{{where}\mspace{14mu}\overset{\sim}{X}}:={\left( {\overset{\sim}{x},\overset{\sim}{\eta}} \right)\mspace{14mu}{and}}}}{{W\left( {\overset{\sim}{X},\sigma,\tau} \right)}:={\begin{pmatrix}{g\left( \overset{\sim}{x} \right)} \\0\end{pmatrix}{{s_{1}\left( {{\alpha\left( {\overset{\sim}{\eta},{H\left( {\overset{\sim}{x},\overset{\sim}{\eta},\sigma,\tau} \right)},\tau} \right)},\sigma} \right)}.}}}} & (13)\end{matrix}$Inverting (13) yields{tilde over (X)}=X−εW(X,σ,εσ)+

_(∞)(ε²).  (14)By lemma 1, this transformation puts (11) into

$\begin{matrix}{{\frac{d\overset{\sim}{X}}{d\;\sigma} = {{ɛ\overset{\_}{\; F}\left( {\overset{\sim}{X},{ɛ\sigma}} \right)} + {ɛ^{2}{\Phi\left( {\overset{\sim}{X},\sigma,{ɛ\sigma}} \right)}} + {\phi\left( {\overset{\sim}{X},\sigma,{ɛ\sigma}} \right)}}};} & (15)\end{matrix}$Φ is periodic and zero-mean in the second argument, and slowly-varyingin average, and ϕ is

_(∞)(ε³) in average.

By lemma 2, the solutions X(σ) and {tilde over (X)}(σ) of (12) and (15),starting from the same initial conditions, satisfy{tilde over (X)}(σ)= X (σ)+

_(∞)(ε³).

As a consequence, the solution X(σ) of (11) stating from X₀ and thesolution X(σ) of (12) starting from X₀−εW(X₀, 0, 0) are related byX(σ)=X(σ)+εW(X(σ), σ, εσ)+

_(∞)(ε²), which is exactly (8a)-(8b). Inserting (8a) in y=h(x) andTaylor expanding yields (8c).

Remark 3: If s₀ were differentiable in the first variable, Φ would beLipschitz and ϕ would be

on a in (15), hence the averaging theory of [9] would directly apply.

Remark 4. In the sequel, we prove for simplicity only the estimation{tilde over (X)}(σ)=X(σ)+

(ε²) on a timescale 1/ε. The continuation to infinity follows from theexponential stability of (4), exactly as in [1, Appendix].

In the same way, lemma 2 is proved without slow-time dependence, thegeneralization being obvious as in [9, section 3.3].

Lemma 1: The transformation (13) puts (11) into (15), where Φ isperiodic and zero-mean in the second argument, and slowly-varying inaverage, ϕ is

_(∞)(ε³) in average.

Proof: To determine the expression for d{tilde over (X)}/dσ, theobjective is to compute dX/dσ as a function of {tilde over (X)} with twodifferent methods. On the one hand we replace X with its transformation(13) in the closed-loop system (11), and on the other hand wedifferentiate (13) with respect to σ.

We first compute s₀(α(η, H(x, η, σ, εσ), εσ), σ) as a function of {tildeover (X)}=({tilde over (x)}, {tilde over (η)}). Exactly as in (10), with({tilde over (x)}, {tilde over (η)}) replacing (x, η), and (14)replacing (9), we haveH (x,η,σ,εσ)=H({tilde over (x)})+

_(∞)(ε²).Therefore, by Taylor expansionα(η, H (x,η,σ,εσ),εσ)=α({tilde over (η)},H({tilde over (x)}),εσ)+ε² K_(α)({tilde over (X)},σ),with K_(α) bounded. The lack of regularity of s₀ prevents further Taylorexpansion; nonetheless, we still can writes ₀(α(η, H (x,η,σ,εσ),εσ),σ)=s ₀(α({tilde over (η)},H({tilde over(x)}),εσ)+ε² K _(α)({tilde over (X)},σ),σ).Finally inserting (13) into (11) and Taylor expanding, yields aftertedious but straightforward computations,

$\begin{matrix}{{\frac{dX}{d\;\sigma} = {{ɛ\;{\overset{\_}{F}\left( {\overset{\sim}{X},{ɛ\sigma}} \right)}} + {ɛ\;{G\left( \overset{\sim}{X} \right)}{s_{0}^{a, +}\left( \overset{\sim}{\cdot} \right)}} + {ɛ^{2}{\overset{\_}{F}\left( {\overset{\sim}{X},{ɛ\sigma}} \right)}{G(X)}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}} + {ɛ^{2}{G^{\prime}\left( \overset{\sim}{X} \right)}{G\left( \overset{\sim}{X} \right)}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}{s_{0}^{a, +}\left( \overset{\sim}{\cdot} \right)}} + {\mathcal{O}_{\infty}\left( ɛ^{3} \right)}}};} & (16)\end{matrix}$we have introduced the following notations

$\left( \overset{\sim}{\cdot} \right):=\left( {\overset{\sim}{X},\sigma,{ɛ\sigma}} \right)$${{s_{i}^{a}\left( \overset{\sim}{\cdot} \right)}:={s_{i}\left( {{\alpha\left( {\overset{\sim}{\eta},{H\left( \overset{\sim}{x} \right)},{ɛ\sigma}} \right)},\sigma} \right)}},{{s_{0}^{a, +}\left( \overset{\sim}{\cdot} \right)}:={s_{0}\left( {{{\alpha\left( {\overset{\sim}{\eta},{H\left( \overset{\sim}{x} \right)},{ɛ\sigma}} \right)} + {ɛ^{2}{K_{a}\left( {\overset{\sim}{X},\sigma} \right)}}},\sigma} \right)}}$${{\Delta s}_{0}^{a}\left( \overset{\sim}{\cdot} \right)}:={{s_{0}^{a, +}\left( \overset{\sim}{\cdot} \right)} - {s_{0}\left( \overset{\sim}{\cdot} \right)}}$${G(X)}:=\begin{pmatrix}{g(x)} \\0\end{pmatrix}$${\overset{\_}{F}\left( {X,{ɛ\sigma}} \right)}:={\begin{pmatrix}{{f(x)} + {{g(x)}{\alpha\left( {\eta,{H(x)},{ɛ\sigma}} \right)}}} \\{\alpha\left( {\eta,{H(x)},{ɛ\sigma}} \right)}\end{pmatrix}.}$

We now time-differentiate (13), which reads with the previous notationsX={tilde over (X)}+εG({tilde over (X)})s ₁ ^(α)({tilde over (·)}).This yields

$\begin{matrix}{{\frac{dX}{d\;\sigma} = {\frac{d\overset{\sim}{X}}{d\;\sigma} + {{{ɛG}^{\prime}\left( \overset{\sim}{X} \right)}\frac{d\overset{\sim}{X}}{d\;\sigma}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}} + {{{ɛG}\left( \overset{\sim}{X} \right)}{\partial_{1}{s_{1}^{\alpha}\left( \overset{\sim}{\cdot} \right)}}\frac{d\overset{\sim}{X}}{d\;\sigma}} + {{{ɛG}\left( \overset{\sim}{X} \right)}{s_{0}^{a}\left( \overset{\sim}{\cdot} \right)}} + {ɛ^{2}{G\left( \overset{\sim}{X} \right)}{\partial_{3}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}}}}},} & (17)\end{matrix}$since ∂₂s₁ ^(α)=s₀ ^(α). Now assume {tilde over (X)} satisfies

$\begin{matrix}{{\frac{d\overset{\sim}{X}}{d\sigma} = {{ɛ{\overset{\_}{F}\left( {\overset{\sim}{X},{ɛ\sigma}} \right)}} + {{{ɛG}\left( \overset{\sim}{X} \right)}{{\Delta s}_{0}^{a}\left( \overset{\sim}{\cdot} \right)}} + {ɛ^{2}{\Psi\left( \overset{\sim}{\cdot} \right)}}}},} & (18)\end{matrix}$where Ψ({tilde over (·)}) is yet to be computed. Inserting (18) into(17),

$\begin{matrix}{\frac{dX}{d\;\sigma} = {{ɛ{\overset{\_}{F}\left( {\overset{\sim}{X},{ɛa}} \right)}} + {{{ɛG}\left( \overset{\sim}{X} \right)}{{\Delta s}_{0}^{a}\left( \overset{\sim}{\cdot} \right)}} + {ɛ^{2}{\Psi\left( \overset{\sim}{\cdot} \right)}} + {ɛ^{2}{G^{\prime}\left( \overset{\sim}{X} \right)}{\overset{\_}{F}\left( {\overset{\sim}{X},{ɛ\sigma}} \right)}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}} + {\epsilon^{2}{G^{\prime}\left( \overset{\sim}{X} \right)}{G\left( \overset{\sim}{X} \right)}{{\Delta s}_{0}^{a}\left( \overset{\sim}{\cdot} \right)}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}} + {ɛ^{2}{G\left( \overset{\sim}{X} \right)}{\partial_{1}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}}{\overset{\_}{F}\left( {\overset{\sim}{X},{ɛ\sigma}} \right)}} + {ɛ^{2}{G\left( \overset{\sim}{X} \right)}{\partial_{1}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}}{G\left( \overset{\sim}{X} \right)}{{\Delta s}_{0}^{a}\left( \overset{\sim}{\cdot} \right)}} + {{{ɛG}\left( \overset{\sim}{X} \right)}{s_{0}^{a}\left( \overset{\sim}{\cdot} \right)}} + {ɛ^{2}{G\left( \overset{\sim}{X} \right)}{\partial_{3}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}}} + {{O_{\infty}\left( ɛ^{3} \right)}.}}} & (19)\end{matrix}$Next, equation (19) and (16), Ψ satisfies

$\begin{matrix}{{\Psi\left( \overset{\sim}{\cdot} \right)} = {{\left\lbrack {\overset{\_}{F},G} \right\rbrack\left( {\overset{\sim}{X},{ɛ\sigma}} \right){s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}} + {{G^{\prime}\left( \overset{\sim}{X} \right)}{G\left( \overset{\sim}{X} \right)}{s_{0}^{a}\left( \overset{\sim}{\cdot} \right)}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}} - {{G\left( \overset{\sim}{X} \right)}{\partial_{1}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}}{\overset{\_}{F}\left( {\overset{\sim}{X},{ɛ\sigma}} \right)}} - {{G\left( \overset{\sim}{X} \right)}{\partial_{3}{s_{1}^{\alpha}\left( \overset{\sim}{\cdot} \right)}}} - {{G\left( \overset{\sim}{X} \right)}{\partial_{1}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}}{G\left( \overset{\sim}{X} \right)}{{{\Delta s}_{0}^{a}\left( \overset{\sim}{\cdot} \right)}.}}}} & (20)\end{matrix}$This gives the expressions of Φ and ϕ in (15),

${{\Phi\left( \overset{\sim}{\cdot} \right)}:={{\left\lbrack {\overset{\_}{F},G} \right\rbrack\left( {\overset{\sim}{X},{ɛ\sigma}} \right){s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}} + {{G^{\prime}\left( \overset{\sim}{X} \right)}{G\left( \overset{\sim}{X} \right)}{s_{0}^{a}\left( \overset{\sim}{\cdot} \right)}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}} - {{G\left( \overset{\sim}{X} \right)}{\partial_{1}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}}{\overset{\_}{F}\left( {\overset{\sim}{X},{ɛ\sigma}} \right)}} - {{G\left( \overset{\sim}{X} \right)}{\partial_{3}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}}}}},\mspace{20mu}{{\phi\left( \overset{\sim}{\cdot} \right)}:={{ɛ^{2}{\Psi_{1}\left( \overset{\sim}{\cdot} \right)}} + {{{ɛG}\left( \overset{\sim}{X} \right)}{\Delta s}_{0}^{a}}}},\mspace{20mu}{with}$$\mspace{20mu}{{\Psi_{1}\left( \overset{\sim}{\cdot} \right)}:={{- {G\left( \overset{\sim}{X} \right)}}{\partial_{1}{s_{1}^{a}\left( \overset{\sim}{\cdot} \right)}}{G\left( \overset{\sim}{X} \right)}{{{\Delta s}_{0}^{a}\left( \overset{\sim}{\cdot} \right)}.}}}$

The last step is to check that Φ and ϕ satisfy the assumptions of thelemma. Since s₀ ^(α), s₁ ^(α), ∂₁s₁ ^(α)and ∂₂s₁ ^(α) are periodic andzero-mean in the second argument, and slowly-varying in average, so isΦ.There remains to prove that ϕ=

_(∞)(ε³) in average. Since Δs₀ ^(α) is slowly-varying in average,∫₀ ^(σ) ∥Δs ₀ ^(α)({tilde over (·)})(s))∥ds≤λ ₀σε².with λ₀>0, G being bounded by a constant c_(g), this implies∥∫₀ ^(σ) εG({tilde over (X)}(s))Δs ₀ ^(α)({tilde over (·)})(s))dsμ≤c_(g)λ₀σε³.Similarly, ∂₁s₁ being bounded by c₁₁, Ψ₁ satisfies∥∫₀ ^(σ)ε²Ψ₁({tilde over (·)}(s))ds∥≤c _(g) ² c ₁₁λ₀σε₀ε³.Summing the two previous inequalities yields∥∫₀ ^(σ)ϕ({tilde over (·)}(s))ds∥≤λ ₀ c _(g)(1+c ₁₁ c _(g)ε₀)σε³,which concludes the proof.

Lemma 2: Let X(σ) and {tilde over (X)}(σ) be respectively the solutionsof (12) and (15) starting at 0 from the same initial conditions. Then,for all σ≥0{tilde over (X)}(σ)= X (σ)+

_(∞)(ε²).

Proof: Let E(σ):={tilde over (X)}(σ)−X(σ). Then,

$\begin{matrix}{{E(\sigma)} = {\int_{0}^{\sigma}{\left\lbrack {{\frac{d\overset{\sim}{X}}{d\sigma}(s)} - {\frac{d\overset{\_}{X}}{d\sigma}(s)}} \right\rbrack{ds}}}} \\{= {{ɛ{\int_{0}^{\sigma}{\left\lbrack {{F\left( {\overset{\sim}{X}(s)} \right)} - {F\left( {\overset{\_}{X}(s)} \right)}} \right\rbrack{ds}}}} + {ɛ^{2}{\int_{0}^{\sigma}{{\Phi\left( {\overset{\sim}{\cdot}(s)} \right)}{ds}}}} +}} \\{\int_{0}^{\sigma}{{\phi\left( {\overset{\sim}{\cdot}(s)} \right)}{ds}}}\end{matrix}$As F Lipschitz with constant λ_(F),ε∫₀ ^(σ) ∥F({tilde over (X)}(s))−F( X (s))∥ds≤ελ _(F)∫₀ ^(σ) ∥E(s)∥ds.On the other hand, there exists by lemma 3 c₁ such thatε²∥∫₀ ^(σ)Φ({tilde over (·)})(s))ds∥≤c ₁ε²Finally, as ϕ is

_(∞)(ε³) in average, there exists c₂ such thatμ∫₀ ^(σ)ϕ({tilde over (·)}(s))ds∥≤c ₂ε³σ.The summation of these estimations yields∥E(σ)∥≤ελ_(F)∫₀ ^(σ) ∥E(s)∥ds+c ₁ε² +c ₂ε³σ.Then by Gronwall's lemma [9, Lemma 1.3.3]

${{{E(\sigma)}} \leq {\left( {\frac{c_{2}}{\lambda_{p}} + c_{1}} \right)e^{\lambda_{F}\sigma}ɛ^{2}}},$which means {tilde over (X)}=X+

_(∞)(ε²).

The following lemma is an extension of Besjes' lemma [9, Lemma 2.8.2]when φ is no longer Lipschitz, but only slowly-varying in average.

Lemma 3: Assume φ(X, σ) is T-periodic and zero-mean in the secondargument, bounded, and slowly-varying in average. Assume the solutionX(σ) of {dot over (X)}=

_(∞)(ε) is defined for 0≤σ≤L/ε. There exists c₁>0 such that∥∫₀ ^(σ)φ(X(s),s)ds∥≤c ₁.

Proof: Along the lines of [9], we divide the interval [0, t] in msubintervals [0, T], . . . , [(m−1)T, mT] and a remainder [mT, t]. Bysplitting the integral on those intervals, we write

${{\int_{0}^{\sigma}{{\varphi\left( {{ϰ(s)},s} \right)}{ds}}} = {{\sum\limits_{i = 0}^{m}\;{\int_{{({i - 1})}T}^{iT}{{\varphi\left( {{ϰ\left( {\left( {i - 1} \right)T} \right)},s} \right)}{ds}}}} + {\sum\limits_{i = 0}^{m}\;{\int_{{({i - 1})}T}^{iT}{\left\lbrack {{\varphi\left( {{ϰ(s)},s} \right)} - {\varphi\left( {{ϰ\left( {\left( {i - 1} \right)T} \right)},s} \right)}} \right\rbrack{ds}}}} + {\int_{mT}^{\sigma}{{\varphi\left( {{ϰ(s)},s} \right)}{ds}}}}},$where each of the integral in the first sum are zero as φ is periodicwith zero mean. Since φ is bounded, the remainder is also bounded by aconstant c₂>0. Besides

x ⁡ ( s ) = x ⁡ ( ( i - 1 ) ⁢ T ) + ∫ ( i - 1 ) ⁢ T ⁢ ϰ ⁢ . ⁢ ( τ ) ⁢ d ⁢ ⁢ τ = ϰ ⁡( ( i - 1 ) ⁢ T ) + ɛ ⁢ q ⁡ ( s ) ,with q continuous and bounded. By hypothesis, there exists λ>0 such that0≤i≤m,∫_((i−)1)T^(T)∥φ(x(s),s)−φ(x((i−1)T),s)∥ds≤λTεTherefore by summing the previous estimations,∥∫₀ ^(σ)φ(x(s),s)ds∥≤mλTε+o ₂,with mT≤t≤L/ε, consequently mλTε+o₂≤λL+o₂; which concludes the proof.

IV. Demodulation

From (8c), we can write the measured signal y as

${{y(t)} = {{y_{a}(t)} + {{y_{v}(t)}{s_{1}\left( {{u(t)},\frac{t}{ɛ}} \right)}} + {O_{\infty}\left( ɛ^{2} \right)}}},$where the signal u feeding the PWM encoder is known. The followingresult shows y_(α) and y_(ν) can he estimated from y, for use in acontrol law as described in section III-A.

Theorem 2: Consider the estimators ŷ_(α) and ŷ_(u) defined by

${{\hat{y}}_{a}(t)}:={{\frac{3}{2}{M(y)}(t)} - {\frac{1}{2}{M(y)}\left( {t - ɛ} \right)}}$${k_{\Delta}(\tau)}:={\left( {{y(\tau)} - {{\hat{y}}_{n}(\tau)}} \right){s_{1}\left( {{u(\tau)},\frac{\tau}{ɛ}} \right)}}$${{{\hat{y}}_{v}(t)}:=\frac{{M\left( k_{\Delta} \right)}(t)}{\overset{\_}{s_{1}^{2}}\left( {u(t)} \right)}},$where M:y

ε⁻¹∫₀ ^(ε)y(τ)dτ is the moving average operator, and s₁ ² the mean of s₁² in the second argument (cf end of section I). Then,ŷ _(α)(t)=y _(α)(t)+

_(∞)(ε²)  (21a)ŷ _(ν)(t)=y _(ν)(t)′

_(∞)(ε²).  (21b)Recall that by construction y_(ν)(t)=

_(∞)(ε), hence (21b) is essentially a first-order estimation; noticealso that s₁ ² (u(t)) is always non-zero when u(t) does not exceed therange of the PWM encoder.

Proof: Taylor expanding y_(α), y_(ν), u and s₁ yields

${y_{a}\left( {t - \tau} \right)} = {{y_{a}(t)} - {r{{\overset{.}{y}}_{a}(t)}} + {O_{\infty}(\tau)}^{2}}$y_(ν)(t − τ) = y_(v)(t) + O_(∞)(ɛ)O_(∞)(τ) $\begin{matrix}{{s_{1}\left( {{u\left( {t - \tau} \right)},\sigma} \right)} = {s_{1}\left( {{{u(t)} + {O_{\infty}(\tau)}},\sigma} \right)}} \\{{= {{s_{1}\left( {{u(t)},\sigma} \right)} + {O_{\infty}(\tau)}}};}\end{matrix}$in the second equation, we have used y_(ν)(t)=

_(∞)(ε). The moving average y_(α) then reads

$\begin{matrix}\begin{matrix}{{{M\left( y_{a} \right)}(t)} = {\frac{1}{ɛ}{\int_{0}^{ɛ}{{y_{a}\left( {t - \tau} \right)}{dr}}}}} \\{= {\frac{1}{ɛ}{\int_{0}^{ɛ}{\left( {{y_{a}(t)} - {\tau{{\overset{.}{\mathcal{y}}}_{a}(t)}} + {O_{\infty}\left( \tau^{2} \right)}} \right)d\;\tau}}}} \\{= {{y_{a}(t)} - {\frac{ɛ}{2}{{\overset{.}{\mathcal{y}}}_{a}(t)}} + {{O_{\infty}\left( ɛ^{2} \right)}.}}}\end{matrix} & (22)\end{matrix}$A similar computation for

${k_{v}(t)}:={{y_{v}(t)}{s_{1}\left( {{u(t)},\frac{t}{ɛ}} \right)}}$yields

$\begin{matrix}\begin{matrix}{{{M\left( k_{\nu} \right)}(t)} = {\frac{1}{ɛ}{\int_{0}^{ɛ}{{y_{\nu}\ \left( {t - \tau} \right)}{s_{1}\left( {{u\left( {t - \tau} \right)},\frac{t - r}{ɛ}} \right)}{d\tau}}}}} \\{= {{{y_{\nu}(t)}\left( {{\overset{\_}{s_{1}}\left( {u(t)} \right)} + {O_{\infty}(ɛ)}} \right)} + {O_{\infty}\left( ɛ^{2} \right)}}} \\{{= {O_{\infty}\left( ɛ^{2} \right)}},}\end{matrix} & (23)\end{matrix}$since s₁ is 1-periodic and zero mean in the second argument. Summing (22and (23), we eventually find

${{M(y)}(t)} = {{y_{a}(t)} - {\frac{ɛ}{2}{{\overset{.}{y}}_{a}(t)}} + {{\mathcal{O}_{\infty}\left( ɛ^{2} \right)}.}}$As a consequence, we get after another Taylor expansion3/2M(y)(t)−½M(y)(t−ε)=y _(α)(t)+

_(∞)(ε²),which is the desired estimation (21a).

On the other hand, (21a) impliesk _(Δ)(t)=y _(ν)(t)s ₁ ²(u(t),t/ε)+

_(∞)(ε²).Proceeding as for M(k_(ν)), we find

$\begin{matrix}{{{M\left( k_{\Delta} \right)}(t)} = {\frac{1}{ɛ}{\int_{0}^{ɛ}{{y_{v}\left( {t - \tau} \right)}{s_{1}^{2}\left( {{u\left( {t - \tau} \right)},\frac{t - \tau}{ɛ}} \right)}d\;\tau}}}} \\{= {{{y_{v}(t)}\left( {{\overset{\_}{s_{1}^{2}}\left( {u(t)} \right)} + {\mathcal{O}_{\infty}(ɛ)}} \right)} + {\mathcal{O}_{\infty}\left( ɛ^{2} \right)}}} \\{= {{{y_{v}(t)}{\overset{\_}{s_{1}^{2}}\left( {u(t)} \right)}} + {{\mathcal{O}_{\infty}\left( ɛ^{2} \right)}.}}}\end{matrix}$Dividing by s₁ ² (u(t)) yields the desired estimation (21b).

V. Numerical Example

We illustrate the interest of the approach on the system{dot over (x)}₁=x₂,{dot over (x)}₂=x₃,{dot over (x)} ₃ =u+d,y=x ₂ +x ₁ x ₃,where d is an unknown disturbance; u will be impressed through PWM withfrequency 1 kHz (i.e. ε=10⁻³) and range [−20, 20]. The objective is tocontrol x₁, while rejecting the disturbance d, with a response time of afew seconds. We want to operate around equilibrium points, which are ofthe form (x₁ ^(eq), 0, 0; −d^(eq), d^(eq)), for x₁ ^(eq) and d^(eq)constant. Notice the observability degenerates at such points, whichrenders not trivial the design of a control law.

Nevertheless the PWM-induced signal injection makes available thevirtual measurement

${y_{v} = {{{ɛ\begin{pmatrix}x_{3} & 1 & x_{1}\end{pmatrix}}\begin{pmatrix}0 \\0 \\1\end{pmatrix}} = {ɛ\; x_{1}}}},$from which it is easy to design a suitable control law, without evenusing the actual input y_(α)=x₂+x₁x₃. The system being now fully linear,we use a classical controller-observer, with disturbance estimation toensure an implicit integral effect. The observes is thus given by

${{\overset{.}{\hat{x}}}_{1} = {{\hat{x}}_{2} + {l_{1}\left( {\frac{y_{v}}{ɛ} - {\hat{x}}_{1}} \right)}}},{{\overset{.}{\hat{x}}}_{2} = {{\hat{x}}_{3} + {l_{2}\left( {\frac{y_{v}}{ɛ} - {\hat{x}}_{1}} \right)}}},{{\overset{.}{\hat{x}}}_{3} = {u + \hat{d} + {l_{3}\left( {\frac{y_{v}}{ɛ} - {\hat{x}}_{1}} \right)}}},{\overset{.}{\hat{d}} = {l_{d}\left( {\frac{y_{v}}{ɛ} - {\hat{x}}_{1}} \right)}},$and the controller byu=−k ₁ {circumflex over (x)} ₁ −k ₂ {circumflex over (x)} ₂ −k ₃{circumflex over (x)} ₃ −k _(d) {circumflex over (d)}+kx ₁ ^(ref).The gains are chosen to place the observer eigenvalues (−1.19, −0.73,−0.49±0.57i) and the controller eigenvalues at (−6.59, −3.30±5.71i). Theobserver is slower than the controller in accordance with dual LoopTransfer Recovery, thus ensuring a reasonable robustness. Settingη:=({circumflex over (x)}₁, {circumflex over (x)}₂, {circumflex over(x)}₃, {circumflex over (d)})^(T), this controller-observer obviouslyreadsu=−Kη+kx ₁ ^(ref)  (24a){dot over (η)}=Mη+Nx ₁ ^(ref)(t)+Ly _(ν)  (24b)Finally, this ideal control law is implemented as

$\begin{matrix}{{u_{pwm}(t)} = {\mathcal{M}\left( {{{{- K}\;\eta} + {kx}_{1}^{ref}},\frac{t}{ɛ}} \right)}} & \left( {25a} \right) \\{{\overset{.}{\eta} = {{M\;\eta} + {Nx}_{1}^{ref} + {L\frac{{\hat{y}}_{v}}{ɛ}}}},} & \left( {25b} \right)\end{matrix}$where M is the PWM function described in section II, and ŷ₈₄ obtained bythe demodulation process of section IV.

The test scenario is the following: at t=0, the system start at rest atthe origin; from t=2, a disturbance d=−0.25 is applied to the system; att=14, a filtered unit step is applied to the reference x₁ ^(ref). InFIG. 5 the ideal control law (24), i.e. without PWM and assuming y_(ν)known, is compared to the true control law (25): the behavior of (25) isexcellent, it is nearly impossible to distinguish the two situations onthe responses of x₁ and x₂ as by (8a) the corresponding ripple is only

_(∞)(ε²); the ripple is visible on x₃, where it is

_(∞)(ε). The corresponding control signals u and u_(pwm) are displayedin FIG. 6, and the corresponding measured outputs in FIG. 7.

To investigate the sensitivity to measurement noise, the same test wascarried out with band-limited white noise (power density 1×10⁻⁹, sampletime 1×10⁻⁵) added to y. Even though the ripple in the measured outputis buried in noise, see FIG. 8, the virtual output is comedy demodulatedand the control law (25) still behaves very well.

CONCLUSION

We have presented a method to take advantage of the benefits of signalinjection in PWM-fed systems without the need for an external probingsignal. For simplicity, have restricted to Single-Input Single-Outputsystems, but then are no essential difficulties to considerMultiple-Input Multiple-Output systems. Besides, though we have focusedon classical PWM, the approach can readily be extended to arbitrarymodulation processes, for instance multilevel PWM; in fact, the onlyrequirements is that s₀ and s₁ meet the regularity assumptions discussedin remark 1.

REFERENCES

-   [1] P. Combes, A. K. Jebai, E Malrait, P. Martin, and P. Rouchon,    “Adding virtual measurements by signal injection,” in American    Control Conference, 2016, pp. 999-1005.-   [2] P. Jansen and R. Lorenz, “Transducerless position and velocity    estimation in induction and salient AC machines,” IEEE Trans.    Industry Applications, vol. 31, pp. 240-247, 1995.-   [3] M. Corley and R. Lorenz, “Rotor position and velocity estimation    for a salient-pole permanent magnet synchronous machine at    standstill and high speeds,” IEEE Trans. Industry Applications, vol.    34, pp. 784-789, 1998.-   [4] A. K. Jebai, E Malrait, P. Martin, and P. Rouchon, “Sensorless    position estimation and control of permanent-magnet synchronous    motors using a saturation model,” International Journal of Control,    vol. 89, no. 3, pp. 535-549, 2016.-   [5] B. Yi, R. Ortega, and W. Zhang, “Relaxing the conditions for    parameter estimation-based observers of nonlinear systems via signal    injection,” Systems and Control Letters, vol. 111, pp. 18-26, 2018.-   [6] C. Wang and L. Xu, “A novel approach for sensorless control of    PM machines down to zero speed without signal injection or special    PWM technique,” IEEE Transactions on Power Electronics, vol. 19, no.    6, pp. 1601-1607, 2004.-   [7] B. Lehman and R. M. Bass, “Extensions of averaging theory for    power electronic systems,” IEEE Transactions on Power Electronics,    vol. 11, no. 4, pp. 542-553, July 1996.-   [8] A. Filippov, Differential equations with discontinuous righthand    sides. Control systems, set Mathematics and its Applications.    Kluwer, 1988.-   [9] J. Sanders, F. Verhulst, and J. Murdock, Averaging methods in    nonlinear dynamical systems, 2nd ed. Springer, 2005.

Sensorless Rotor Position Estimation by PWM-Induced Signal Injection

Abstract: We demonstrate how the rotor position of a PWM-controlled PMSMcan be recovered from the measured currents, by suitably using theexcitation provided by the PWM itself. This provides the benefits ofsignal injection, in particular the ability to operate even at lowvelocity, without the drawbacks of an external probing signal. Weillustrate the relevance of the approach by simulations and experimentalresults.

Index Terms—Sensorless control, PMSM, signal injection, PWM-inducedripple.

Nomenclature

-   PWM Pulse Width Modulation-   x^(dq) Vector (x^(d), x^(q))^(T) in the dq frame-   x^(αβ) Vector (x^(α), x^(β))^(T) in the αβ frame-   x^(abc) Vector (x^(a), x^(b), x^(c)) in the abc frame-   R_(s) Stator resistance-   Rotation matrix with angle π/2;

$\quad\begin{pmatrix}0 & {- 1} \\1 & 0\end{pmatrix}$

-   I Moment of inertia-   n Number of pole pairs-   ω Rotor speed-   T_(l) Load torque-   θ, {circumflex over (θ)} Actual, estimated rotor position-   ϕ_(m) Permanent magnet flux-   L_(d), L_(q) d and q-axis inductances-   C Clarke transformation:

$\frac{2}{3}\begin{pmatrix}1 & {{- 1}/2} & {{- 1}/2} \\0 & {\sqrt{3}/2} & {{- \sqrt{3}}/2}\end{pmatrix}$

-   (θ) Rotation matrix with angle θ:

$\quad\begin{pmatrix}{\cos\mspace{11mu}\theta} & {{- \sin}\mspace{11mu}\theta} \\{\sin\mspace{11mu}\theta} & {\cos\mspace{11mu}\theta}\end{pmatrix}$

-   ε PWM period-   u_(m) PWM amplitude-   S(θ) Saliency matrix-   O “Big O” symbol of analysis: k(z, ε)=O(ε) means ∥k(z, ε)∥≤Cε, for    some C independent of z and ε.

I. Introduction

Sensorless control of AC motors in the low-speed range is a challengingtask. Indeed, the observability of the system from the measurements ofthe currents degenerates at standstill, which limits the performance atlow speed of any fundamental-model-based control law.

One now with widespread method to overcome this issue is the so-calledsignal injection technique. It consists in superimposing a fast-varyingsignal to the control law. This injection creates ripple on the currentmeasurements which carries information on the rotor position if properlydecoded. Nonetheless, introducing a fast-varying signal increasesacoustic noise and may excite mechanical resonances. For systemscontrolled through Pulse Width Modulation (PWM), the injection frequencyin moreover inherently limited by the modulation frequency. That said,inverter-friendly waveforms can also be injected to produce the sameeffect, as in the so-called INFORM method [1], [2]. For PWM-fedPermanent Magnet Synchronous Motors (PMSM), the oscillatory nature ofthe input may be seen as a kind of generalised rectangular injection onthe three input voltages, which provides the benefits of signalinjection, in particular the ability to operate even at low velocity,without the drawbacks of an external probing signal.

We build on the quantitative analysis developed (3) to demonstrate howthe rotor position of a PWM-controlled PMSM can be recovered from themeasured currents, by suitably using the excitation provided by the PWMitself. No modification of the PWM stage nor injection a high-frequencysignal as in [4] is required.

The paper runs as follows: we describe in section II the effect of PWMon the current measurements along the lines of [3], slightlygeneralizing to the multiple-input multiple output framework. In sectionIII, we show how the rotor position can be recovered for two PWM schemesschemes, namely standard single-carrier PWM and interleaved PWM. Therelevance of the approach is illustrated in section IV with numericaland experimental results.

II. Virtual Measurement Induced by PWM

Consider the state-space model of a PMSM in the dq frame

$\begin{matrix}{{\frac{d\;\phi_{s}^{dq}}{dt} = {u_{x}^{dq} - {R_{s}l_{s}^{dq}} - {\omega\mathcal{J}\phi}_{s}^{dq}}},} & \left( {1a} \right) \\{{{\frac{J}{n}\frac{d\omega}{dt}} = {{w_{x}^{{dq}^{\tau}}{\mathcal{J}\phi}_{s}^{dq}} - T_{l}}},} & \left( {1b} \right) \\{{\frac{d\;\theta}{d\; t} = \omega},} & \left( {1c} \right)\end{matrix}$where ϕ_(s) ^(dq) is the stator flux linkage, ω the rotor speed, θ therotor position, t_(s) ^(dq) the stator current, u_(s) ^(dq) the statorvoltage, and T_(l) the load torque; R_(s), J, and n are constantparameters (see nomenclature for notations). For simplicity we assume nomagnetic saturation, i.e. linear current-flux relationsL _(d) _(l) _(s) ^(d)=ϕ_(s) ^(d)−ϕ_(m)  (2a)L_(q) _(l) _(s) ^(q)=ϕ_(s) ^(q),  (2b)with ϕ_(m) the permanent magnet flux; see [5] for a detailed discussionof magnetic saturation in the context of signal injection. The input isthe voltage u_(s) ^(abc) through the relationu _(s) ^(dq)=

(−θ)Cu _(s) ^(abc).   (3)In an industrial drive, the voltage actually impressed is not directlyu_(s) ^(abc), but its PWM encoding

${\mathcal{M}\left( {u_{s}^{abc},\frac{t}{ɛ}} \right)},$with ε the PWM period. The function

describing the PWM is 1-periodic and mean u_(s) ^(abc) in the secondargument, i.e.

(u_(s) ^(abc), τ+1)=

(u_(s) ^(abc), τ) and ∫₀ ¹

(u_(s) ^(abc), τ)dτ=u_(s) ^(abc); its expression is given in sectionIII. Setting s₀ ^(abc)(u_(s) ^(abc), σ):=(

(u_(s) ^(abc), σ)−u_(s) ^(abc), the impressed voltage thus reads

${u_{pwm}^{abc} = {u_{s}^{abc} + {s_{0}^{abc}\left( {u_{s}^{abc},\frac{t}{ɛ}} \right)}}},$where s₀ ^(abc) is 1-periodic and zero mean in the second argument; s₀^(abc) can be seen as a PWM-induced rectangular probing signal, whichcreates ripple but has otherwise no effect. Finally, as we are concernedwith sensorless control, the only measurement is the current i_(s)^(abc)=C^(T)

(θ)_(l) _(s) ^(dq), or equivalently i_(s) ^(αβ)=

(θ)_(l) _(s) ^(dq) since i_(s) ^(a)+i_(s) ^(b)+i_(s) ^(c)=0.

A precise quantitative analysis of signal injection is developed in [3],[6]. Slightly generalizing these results to the multiple-inputmultiple-output case, the effect of PWM-induced signal injection can beanalyzed thanks to second-order averaging in the following way. Considerthe system

${\overset{.}{x} = {{f(x)} + {{g(x)}\left( {u + {s_{0}\left( {u,\frac{t}{ɛ}} \right)}} \right)}}},{y = {h(x)}},$where u is the control input, ε is a the (assumed small) PWM period, ands₀ is 1-periodic in the second argument, with zero mean in the secondargument; then we can extract from the actual measurement y with anaccuracy of order ε the so-called virtual measurement (see [3], [6])y _(v)(t):=h′(x(t))g(x(t))

(u(t)),i.e. we can compute by a suitable filtering process an estimate

(t)=y _(v)(t)+O(ε).The matrix

, which can be computed online, is defined by

(ν):=∫₀ ¹ s ₁(ν,τ)s ₁ ^(T)(ν,τ)dτ,where s₁ is the zero-mean primitive in the second argument of s₀, i.e.s ₁(ν,τ):=∫₀ ¹ s ₀(ν,τ)dσ−∫ ₀ ¹∫₀ ^(τ) s ₀(ν,σ)dσdτ.The quantity

$ɛ\;{h^{\prime}\left( {x(t)} \right)}{g\left( {x(t)} \right)}{s_{1}\left( {{u(t)},\frac{t}{ɛ}} \right)}$is the ripple caused on the output y by the excitation signal

${s_{0}\left( {{u(t)},\frac{t}{ɛ}} \right)};$though small, it contains valuable information when properly processed.

For the PMSM (1)-(3) with output i_(s) ^(αβ), some algebra yields

${y_{v} = {{{\begin{bmatrix}{{\mathcal{R}(\theta)}\begin{pmatrix}\frac{1}{L_{d}} & 0 \\0 & \frac{1}{L_{q}}\end{pmatrix}} & 0_{2 \times 1} & {{\mathcal{R}^{\prime}(\theta)}t_{dq}}\end{bmatrix}\begin{bmatrix}{{\mathcal{R}\left( {- \theta} \right)}\mathcal{C}} \\0_{1 \times 2} \\0_{1 \times 2}\end{bmatrix}}{\mathcal{A}^{abc}\left( u^{abc} \right)}} = {{\mathcal{S}(\theta)}{{\mathcal{C}\mathcal{A}}^{abc}\left( u^{abc} \right)}}}},\mspace{20mu}{where}$  𝒜^(abc)(v^(abc)) := ∫₀¹s₁^(abc)(v^(abc), τ)s₁^(abc^(τ))(v^(abc), τ)d τ,and S(θ) is the co-called saliency matrix introduced in [5],

${\mathcal{S}(\theta)} = {\frac{L_{d} + L_{q}}{2L_{d}L_{q}}{\begin{pmatrix}{1 + {\frac{L_{q} - L_{d}}{L_{d} + L_{q}}\cos\mspace{11mu} 2\theta}} & {\frac{L_{q} - L_{d}}{L_{d} + L_{q}}\sin\mspace{11mu} 2\theta} \\{\frac{L_{q} - L_{d}}{L_{d} + L_{q}}\sin\mspace{11mu} 2\theta} & {1 - {\frac{L_{q} - L_{d}}{L_{d} + L_{q}}\cos\mspace{11mu} 2\theta}}\end{pmatrix}.}}$If the motor has sufficient geometric saliency, i.e. if L_(d) and L_(q)are sufficiently different, the rotor position θ can be extracted fromy_(v) as explained in section III. When geometric saliency is small,information on θ is usually still present when magnetic saturation istaken into account, see [5].

III. Extracting θ from the Virtual Measurement

Extracting the rotor position θ from y_(v) depends on the rank of the2×3 matrix C

^(abc)(ν^(abc)). The structure of this matrix, hence its rank, dependson the specifies of the PWM employed. After recalling the basics ofsingle-phase PWM, we study two cases: standard three-phase PWM with asingle carrier, and three-phase PWM with interleaved carriers.

Before that, we notice that C

^(abc)(ν^(abc)) has the same rank as the 2×2 matrix

^(αβ)(ν^(abc)):=C

^(abc)(ν^(abc))C ^(T)=∫₀ ¹ s ₁ ^(αβ)(ν^(abc),τ)s ₁ ^(αβ) ^(T)(ν^(abc),τ)dτ,where s₁ ^(αβ)(ν^(abc), τ):=Cs₁ ^(abc)(ν^(abc), τ). Indeed,

^(αβ)(ν^(abc))

^(αβ) ^(T) (ν^(abc))=C

^(abc)(ν^(abc))C ^(T) C

^(abc) ^(T) (ν^(abc))C ^(T) =C

^(abc)(ν^(abc))(C

^(abc)(ν^(abc)))^(T),which means that

^(αβ)(ν^(abc) and C

^(abc)(ν^(abc)) have the same singular values, hence the same rank.There is thus no loss of information when considering S(θ)

^(αβ)(u^(abc)) instead of the original virtual measurement y_(v).A. Single-Phase PWM

In “natural” PWM with period ε and range [−u_(m), u_(m)], the inputsignal u is compared to the ε-periodic triangular carrier

${c(t)}:=\left\{ \begin{matrix}{u_{m} + {4{w\left( \frac{t}{ɛ} \right)}}} & {if} & {{- \frac{u_{m}}{2}} \leq {w\left( \frac{t}{ɛ} \right)} \leq 0} \\{u_{m} - {4{w\left( \frac{t}{ɛ} \right)}}} & {if} & {0 \leq {w\left( \frac{t}{ɛ} \right)} \leq \frac{u_{m}}{2}}\end{matrix} \right.$the 1-periodic function

${w(\sigma)}:={{u_{m}{{mod}\left( {{\sigma + \frac{1}{2}},1} \right)}} - \frac{u_{m}}{2}}$wraps the normalized time

$\sigma = {\frac{t}{ɛ}\mspace{14mu}{{{to}\mspace{14mu}\left\lbrack {{- \frac{u_{m}}{2}},\frac{u_{m}}{2}} \right\rbrack}.}}$If u varies slowly enough, it crosses the carrier c exactly once on eachrising and falling ramp, at times t₁ ^(u)<t₂ ^(u) such that

${u\left( t_{1}^{u} \right)} = {u_{m} + {4{w\left( \frac{t_{1}^{u}}{ɛ} \right)}}}$${u\left( t_{2}^{u} \right)} = {u_{m} - {4{{w\left( \frac{t_{2}^{u}}{ɛ} \right)}.}}}$The PWM-encoded signal is therefore given by

${u_{pwm}(t)} = \left\{ {\begin{matrix}u_{m} & {if} & {{- \frac{u_{m}}{2}} < {W\left( \frac{t}{ɛ} \right)} \leq {W\left( \frac{t_{1}^{u}}{ɛ} \right)}} \\{- u_{m}} & {if} & {{W\left( \frac{t_{1}^{u}}{ɛ} \right)} < {W\left( \frac{t}{ɛ} \right)} \leq \left( \frac{t_{2}^{u}}{ɛ} \right)} \\u_{m} & {if} & {{W\left( \frac{t_{2}^{u}}{ɛ} \right)} < {W\left( \frac{t}{ɛ} \right)} \leq \frac{u_{m}}{2}}\end{matrix}.} \right.$FIG. 10 illustrates the signals u, c and u_(pwm). The function

$\begin{matrix}{{\mathcal{M}\left( {u,\sigma} \right)}:=\left\{ \begin{matrix}u_{m} & {if} & {{{- 2}u_{m}} < {4{w(\sigma)}} \leq {u - u_{m}}} \\{- u_{m}} & {if} & {{u - u_{m}} < {4{w(\sigma)}} \leq {u_{m} - u}} \\u_{m} & {if} & {{u_{m} - u} < {4{w(\sigma)}} \leq {2u_{m}\overset{\square}{\square}}}\end{matrix} \right.} \\{= {u_{m} + {u_{m}\mspace{14mu}{{sign}\left( {u - u_{m} - {4{w(\sigma)}}} \right)}} +}} \\{{u_{m}\mspace{14mu}{{sign}\left( {u - u_{m} + {4{w(\sigma)}}} \right)}},}\end{matrix}$which is obviously 1-periodic and with mean u with respect to its secondargument, therefore completely describes the PWM process since

${u_{pwn}(t)} = {{\mathcal{M}\left( {{u(t)},\frac{t}{ɛ}} \right)}.}$

The induced zero-mean probing signal is then

${{s_{0}\left( {u,\sigma} \right)}:={{{\mathcal{M}\left( {u,\sigma} \right)} - u} = {u_{m} - u + {u_{m}\mspace{14mu}{{sign}\left( {\frac{u - u_{m}}{4} - {w(\sigma)}} \right)}} + {u_{m}\mspace{14mu}{{sign}\left( {\frac{u - u_{m}}{4} + {W(\sigma)}} \right)}}}}},$and its zero-mean primitive in the second argument is

${s_{1}\left( {u,\sigma} \right)}:={{\left( {1 - \frac{u}{u_{m}}} \right){w(\sigma)}} - {{\frac{u - u_{m}}{4} - {w(\sigma)}}} + {{{\frac{u - u_{m}}{4} + {w(\sigma)}}}.}}$The signals s₀, s₁ and w are displayed in FIG. 11. Notice that byconstruction s₀(±u_(m), σ)=s₁(±u_(m), σ)=0, so there is no ripple, henceno usable information, at the PWM limits.B. Three-Phase PWM with Single Carrier

In three-phase PWM with single carrier, each component u_(s) ^(k), k=a,b, c, of u_(x) ^(abc) is compared to the same carrier, yieldings ₀ ^(k)(u _(s) ^(abc),σ):=s ₀(u _(s) ^(k),σ)s ₁ ^(k)(u _(s) ^(abc),σ):=s ₁(u _(s) ^(k),σ),with s₀ and s₁ as in single-phase PWM. This is the most common PWM inindustrial drives as it is easy to implement.

Notice that if exactly two components of u_(s) ^(abc) are equal, forinstance u_(s) ^(c)=u_(s) ^(b)≠u_(s) ^(a), thens ₁ ^(c)(u _(s) ^(abc),σ)=s ₁ ^(b)(u _(s) ^(abc),σ)≠s ₁ ^(a)(u _(s)^(abc),σ),which implies in turn that

^(αβ)(u^(abc)) has rank 1 (its determinant vanishes); it can be shownthis is the only situation that results in rank 1. If all three commentsof u_(s) ^(abc) are equal, then

^(αβ)(u^(abc)) has rank 0 (i.e. all its entries are zero); this is arather exceptional condition that we rule out here. Otherwise

^(αβ)(u^(abc)) has rank 2 (i.e. is invertible). FIG. 12 displaysexamples of the shape of s₁ ^(αβ), in the rank 2 case (top), and in therank 1 case where u_(s) ^(c)=u_(s) ^(b)≠u_(s) ^(a).

As the rank 1 situation very often occurs, it must be handled by theprocedure for extracting θ from S(θ)

^(αβ)(u^(abc)). This can be done by linear least squares, thanks to theparticular structure of S(θ). Setting

$\begin{pmatrix}\lambda & \mu \\\mu & v\end{pmatrix}:={\mathcal{A}^{\alpha\;\beta}\left( u^{abc} \right)}$$\begin{pmatrix}y_{11} & y_{12} \\y_{21} & y_{22}\end{pmatrix}:={\frac{2L_{d}L_{q}}{L_{d} + L_{q}}{y_{v}.}}$and

${L:=\frac{L_{d} + L_{q}}{L_{q} - L_{d}}},$we can rewrite y_(v) =S(θ)

^(αβ)(u^(abc)) as

${\underset{\underset{:=P}{︸}}{\begin{pmatrix}\lambda & \mu \\\mu & v \\{- \mu} & \lambda \\{- v} & \mu\end{pmatrix}}\begin{pmatrix}{\cos\; 2\theta} \\{\sin\; 2\;\theta}\end{pmatrix}} = {L{\underset{\underset{:=d}{︸}}{\begin{pmatrix}{y_{11} - \lambda} \\{y_{12} - \mu} \\{y_{21} - \mu} \\{y_{22} - v}\end{pmatrix}}.}}$The least-square solution of this (consistent) overdetermined linearsystem is

${{rCl}\begin{pmatrix}{\cos\; 2\theta} \\{\sin\; 2\;\theta}\end{pmatrix}} = {{{L\left\lbrack {P^{T}P} \right\rbrack}^{- 1}P^{T}d} = {\frac{L}{\lambda^{2} + {2\mu^{2}} + v^{2}}P^{T}}}$$d = {\frac{L}{\lambda^{2} + {2\mu^{2}} + v^{2}}{\begin{pmatrix}{{\lambda\; y_{11}} + {\mu\left( {y_{12} - y_{21}} \right)} - {vy}_{22} - \lambda^{2} + v^{2}} \\{{\lambda\;\left( {y_{11} + y_{22}} \right)} + {vy}_{12} + {\lambda\; y_{21}} - {2{\mu\left( {\lambda + v} \right)}}}\end{pmatrix}.}}$

Estimates

,

for cos 2θ, sin 2θ are obtained with the same formulas, using instead ofthe actual y_(ij) the estimated

$\begin{pmatrix} & \\ & \end{pmatrix}:={{\frac{2L_{d}L_{q}}{L_{d} + L_{q}}} = {{\frac{2L_{d}L_{q}}{L_{d} + L_{q}}y_{v}} + {{\mathcal{O}(ɛ)}.}}}$We thus have

$:={{L\frac{+ {\mu( - )} - {v} - \lambda^{2} + {v^{2}}}{\lambda^{2} + {2\mu^{2}} + v^{2}}} = {{\cos\; 2\theta} + {\mathcal{O}(ɛ)}}}$$:={{L\frac{{\mu( + )} + {v} + {\lambda} - {2{\mu\left( {\lambda + v} \right)}}}{\lambda^{2} + {2\mu^{2}} + v^{2}}} = {{\sin\; 2\theta} + {{\mathcal{O}(ɛ)}.}}}$Finally, we get an estimate {circumflex over (θ)} of θ by{circumflex over (θ)}:=½ atan 2(

,

)+kπ=θ+O(ε),where kε

is the number of turns.C. Three-Phase PWM with Interleaved Carriers

At the cost of a more complicated implementation, it turns out that aPWM scheme with (regularly) interleaved carries offers several benefitsover single-carrier PWM. In this scheme, each component of u_(s) ^(abc)is compared to a shifted version of the same triangular carrier (withshift 0 for axis a, ⅓ for axis b, and ⅔ axis c), yielding

$\begin{matrix}{{s_{0}^{a}\left( {u_{s}^{abc},\sigma} \right)}:={s_{0}\left( {u_{s}^{a},\sigma} \right)}} & {{s_{1}^{a}\left( {u_{s}^{abc},\sigma} \right)}:={s_{1}\left( {u_{s}^{a},\sigma} \right)}} \\{{s_{0}^{b}\left( {u_{s}^{abc},\sigma} \right)}:={s_{0}\left( {u_{s}^{b},{\sigma - \frac{1}{3}}} \right)}} & {{s_{1}^{b}\left( {u_{s}^{abc},\sigma} \right)}:={s_{1}\left( {u_{s}^{b},{\sigma - \frac{1}{3}}} \right)}} \\{{s_{0}^{c}\left( {u_{s}^{abc},\sigma} \right)}:={s_{0}\left( {u_{s}^{c},{\sigma - \frac{2}{3}}} \right)}} & {{s_{1}^{c}\left( {u_{s}^{abc},\sigma} \right)}:={{s_{1}\left( {u_{s}^{c},{\sigma - \frac{2}{3}}} \right)}.}}\end{matrix}$FIG. 13 illustrates the principle of this scheme. FIG. 14 displays anexample of the shape of s₁ ^(αβ), which always more or less looks liketwo signals in quadrature.

Now, even when two, or even three, components of u_(s) ^(abc) are equal,

^(αβ)(u^(abc)) remains invertible (except of course at the PWM limits),since each component has, because of the interleaving, a different PWMpattern. It is therefore possible to recover all four entries of thesaliency matrix S(θ) by

:=

·[

^(αβ)(u ^(abc))]⁻¹ =S(θ)+O(ε).Notice now that thanks to the structure of S(θ)=(s_(ij))_(ij), the rotorangle θ be computed from the matrix entries by

${s_{12} + s_{21}} = {\frac{L_{q} - L_{d}}{L_{d}L_{q}}\sin\; 2\;\theta}$${s_{11} - s_{22}} = {\frac{L_{q} - L_{d}}{L_{d}L_{q}}\cos\; 2\;\theta}$${\theta = {{\frac{1}{2}{atan}\; 2\left( {{s_{12} + s_{21}},{s_{11} - s_{22}}} \right)} + {k\;\pi}}},$θ=½ atan2(s ₁₂+s ₂₁,s ₁₁−s ₂₂)+kπ,where kε

is the number of turns. An estimate {circumflex over (θ)} of θ cantherefore be computed from the entries (

)_(ij) of

by{circumflex over (θ)}=½ atan 2(

+

,

−

)+kπ=θ+O(ε),without requiring the knowledge the magnetic parameters L_(d) and L_(q),which is indeed a nice practical feature.

IV. Simulations and Experimental Results

The demodulation procedure is tested both in simulation andexperimentally. All the tests, numerical and experimental, use therather salient PMSM with parameters listed in Table 1. The PWM frequencyis 4 kHz.

The test scenario is the following: starting from rest at t=0 s, themotor remains there for 0.5 s, then follows a velocity ramp from 0 to 5Hz (electrical), and finally stays at 5 Hz from t=8.5 s; during all theexperiment, it undergoes a constant load torque of about 40% of therated torque. As this paper is only concerned with the estimation of therotor angle θ, the control law driving the motor is allowed to use themeasured angle. Besides, we are not yet able to process the data inreal-time, hence the data are recorded and processed offline.

TABLE 1 RATED PARAMETERS Rated power 400 W Rated voltage (RMS) 400 VRated current (RMS) 1.66 A Rated speed 1800 RPM Rated torque 2.12 N.mNumber of pole pairs n 2 Moment of inertia J 6.28 kg.cm² Statorresistance R_(a) 3.725 Ω d-axis inductance L_(d) 33.78 mH q-axisinductance L_(q) 59.68 mHA. Single Carrier PWM.

The results obtained in simulation by the reconstruction procedure ofsection III-B for cos 2θ, sin 2θ, and θ, are shown in FIG. 15 and FIG.16. The agreement between the estimates and the actual values isexcellent.

The corresponding results on experimental data are shown in FIG. 19 andFIG. 20. Though of course not as good as in simulation, the agreementbetween the estimates and the ground truth is still very satisfying. Theinfluence of magnetic saturation may account for part of thediscrepancies.

FIG. 17 displays a close view of the ripple envelope s₁ ^(αβ) inapproximately the same conditions as in FIG. 12 when the rank of

^(αβ)(y^(abc)) is 2 case (top) and when the rank is 1 case with u_(s)^(c)=u_(s) ^(b)≠u_(s) ^(a). They illustrate that though the experimentalsignals are distorted, they are nevertheless usable for demodulation.

Finally, we point out an important difference between the simulation andexperimental data. In the experimental measurements, we notice periodicspikes in the current measurement, see FIG. 18; these are due to thedischarges of the parasitic capacitors in the inverter transistors eachtime a PWM commutation occurs. As it might hinder the demodulationprocedure of [3], [6], the measured currents were first preprocessed bya zero-phase (non-casual) moving average with a short window length of0.01ε. We are currently working on an improved demodulation procedurenot requiring prefiltering.

B. Interleaved PWM (Simulation)

The results obtained in simulation by the reconstruction procedure ofsection III-C for the saliency matrix and S(θ) and for θ are shown inFIG. 21 and FIG. 22. The agreement between the estimates and the actualvalues is excellent. We insist that the reconstruction does not requirethe knowledge of the magnetic parameters.

V. Conclusion

This paper provides an analytic approach for the extraction of the rotorposition of a PWM-fed PSMM, with signal injection provided by the PWMitself. Experimental and simulations results illustrate theeffectiveness of this technique.

Further work includes a demodulation strategy not requiring prefilteringof the measured currents, and suitable for real-time processing. Theultimate goal is of course to be able to use the estimated rotorposition inside a feedback loop.

REFERENCES

-   [1] M. Schroedl, “Sensorless control of ac machines at low speed and    standstill based on the “inform” method,” in IAS '96. Conference    Record of the 1996 IEEE Industry Applications Conference    Thirty-First IAS Annual Meeting, vol. 1, 1996, pp. 270-277 vol. I.-   [2] E. Robeischl and M. Schroedl, “Optimized inform measurement    sequence for sensorless pm synchronous motor drives with respect to    minimum current distortion,” IEEE Transactions on Industry    Applications, vol. 40, no. 2, 2004.-   [3] D. Surroop, P. Combes, P. Martin, and P. Rouchon, “Adding    virtual measurements by pwm-induced signal injection,” in 2020    American Control Conference (ACC), 2020, pp. 2692-2698.-   [4] C. Wang and L. Xu, “A novel approach for sensorless control of    PM machines down to zero speed without signal injection or special    PWM technique,” IEEE Transactions on Power Electronics, vol. 19, no.    6, pp. 1601-1607, 2004.-   [5] A. K. Jebai, F. Malrait, P. Martin, and P. Rouchon, “Sensorless    position estimation and control of permanent-magnet synchronous    motors using a saturation model,” International Journal of Control,    vol. 89, no. 3, pp. 535-549, 2016.-   [6] P. Combes, A. K. Jebai, F. Malrait, P. Martin, and P. Rouchon,    “Adding virtual measurements by signal injection,” in American    Control Conference, 2016, pp. 999-1005.

The present disclosure also relates to the following subject-matter:

-   Clause 1: A method for controlling an actuator, comprising providing    a control signal for the actuator and modulating the control signal    by a modulation signal.-   Clause 2: The method of clause 1, further comprising determining a    zero-mean primitive of the modulated control signal, obtaining    measurement signals from the actuator, obtaining virtual    measurements by demodulating the measurement signals based on the    zero-mean primitive of the modulated control signal, adapting the    control signal in response to the obtained virtual measurements.-   Clause 3: The method of clause 1 or 2, wherein the modulation signal    comprises pulse-width modulation.-   Clause 4: The method of any one of clauses 1 to 3, wherein the    actuator is an electric motor.-   Clause 5: A control system for controlling an actuator, comprising a    control module generating a control signal, a PWM-module generating    a modulated control signal by pulse-width modulation and a zero-mean    primitive of the modulated control signal, a measurement unit for    obtaining measurement signals from the actuator, and an estimator    module for obtaining virtual measurements by demodulating the    measurement signals based on the zero-mean primitive of the    modulated control signal, wherein the control module is arranged for    adapting the control signal based on feedback in the form of the    obtained virtual measurements.-   Clause 6: The system of clause 5, wherein the modulation signal    comprises pulse-width modulation.-   Clause 7: The system of clause 5 or 6, wherein the actuator is an    electric motor.

This disclosure is not limited to the specific embodiments describedhere, which are only examples. The invention encompasses everyalternative that a person skilled in the art would envisage when readingthis text.

The invention claimed is:
 1. A variable speed drive for the closed loopcontrol of the operation of an AC electric motor based on a givencontrol law, the variable speed drive comprising: an output terminal fordelivering a controlled alternating drive voltage (upwm) to thecontrolled AC electric motor; a solid-state power inverter forgenerating the drive voltage; a drive controller for controlling thegeneration of the drive voltage by the power inverter; and a drivecurrent sensing device for measuring the instantaneous intensity of thedrive current taken up by the controlled AC electric motor and forproviding the resulting measurements as a drive current intensity signalto the drive controller, wherein the drive controller includes: apulse-width modulation generator; a control law module storing the givencontrol law; and a state variable estimation module for estimating theinstantaneous value of at least one state variable of the controlled ACelectric motor, wherein the control law module is adapted to, based onthe stored control law and state variable estimates provided by theestimation module, compute a target voltage signal and output thecomputed target voltage signal to the pulse-width modulation generator,wherein the pulse-width modulation generator is adapted to: approximatethe received target voltage signal with a pulse-width modulated invertercontrol signal; control the operation of the power inverter using theinverter control signal, thereby obtaining the drive voltage; compute,based on the deviation between the inverter control signal and thetarget voltage signal, a state variable estimation support signal; andoutput the computed state variable estimation support signal to thestate variable estimation module, and wherein the state variableestimation module is adapted to: estimate the instantaneous value of astate variable of the AC electric motor based on the received statevariable estimation support signal and the drive current intensitysignal provided by the drive current sensing device; and output theresulting state variable estimate to the control law module.
 2. Thevariable speed drive of claim 1, wherein the pulse-width modulationgenerator is adapted to compute the state variable estimation supportsignal based on a pulse-width modulation inherent disturbance signal,which is obtained by subtracting the target voltage signal from theinverter control signal.
 3. The variable speed drive of claim 2, whereinthe pulse-width modulation generator is adapted to compute the statevariable estimation support signal by integrating the disturbance signalto obtain the primitive of the disturbance signal.
 4. The variable speeddrive of claim 1, wherein the variable speed drive is adapted to rely ona single feedback to perform closed loop control of the AC electricmotor, namely the drive current intensity signal provided by the drivecurrent sensing device.
 5. The variable speed drive of claim 1, whereinthe variable speed drive is adapted to control the operation of the ACelectric motor without the injection of a dedicated probing signal intothe drive voltage.
 6. The variable speed drive of claim 1, wherein thedrive controller further includes an analog-to-digital converter forconverting the drive current intensity signal into a digital signalprior to its input into the state variable estimation module.
 7. Thevariable speed drive of claim 1, wherein the state variable estimationmodule is adapted to estimate the instantaneous value of the rotorposition of the electric motor based on the received state variableestimation support signal and the drive current intensity signalprovided by the drive current sensing device.
 8. The variable speeddrive of claim 1, wherein the pulse-width modulation generator isadapted to apply three-phase pulse-width modulation with single carrierto generate the inverter control signal.
 9. The variable speed drive ofclaim 1, wherein the pulse-width modulation generator is adapted toapply three-phase pulse-width modulation with interleaved carriers togenerate the inverter control signal.
 10. An electric drive assemblycomprising a synchronous reluctance motor and a variable speed drive ofclaim 1 for controlling the synchronous reluctance motor.
 11. Anelectric drive assembly comprising a permanent-magnet synchronous motorand a variable speed drive of claim 1 for controlling thepermanent-magnet synchronous motor.
 12. A method of controlling, in aclosed loop, the operation of an AC electric motor based on a givencontrol law, the method comprising the following steps: a) measuring theinstantaneous intensity of the drive current taken up by the controlledAC electric motor; b) estimating the instantaneous value of a statevariable of the AC electric motor using the measured drive currentintensity; c) computing, based on the given control law and theestimated state variable, a target voltage signal; d) approximating thecomputed target voltage signal with a pulse-width modulated invertingcontrol signal; e) computing, based on the deviation between theinverting control signal and the target voltage signal, a state variableestimation support signal; f) generating, by voltage inversion, acontrolled alternating drive voltage using the inverting control signal;and g) delivering the generated drive voltage to the controlled ACelectric motor; wherein the state variable estimation according to stepb) relies on the state variable estimation support signal computed instep e) as an additional input together with the drive current intensitymeasured in step a).